Magic Squares and Syzygies Richard P. Stanley
|
|
|
- Tracy Blair
- 7 years ago
- Views:
Transcription
1 Magic Squares and Syzygies p. 1 Magic Squares and Syzygies Richard P. Stanley
2 Magic Squares and Syzygies p. 2 Traditional magic squares Many elegant, ingenious, and beautiful constructions, but no general theory involving all of them.
3 Magic Squares and Syzygies p. 3 Magic squares of the third kind... Definition. An n n magic square with line sum r is an n n matrix of nonnegative integers for which every row and column sums to r.
4 Magic Squares and Syzygies p. 3 Magic squares of the third kind... Definition. An n n magic square with line sum r is an n n matrix of nonnegative integers for which every row and column sums to r n = 3, r = 7
5 Magic Squares and Syzygies p. 3 Magic squares of the third kind... Definition. An n n magic square with line sum r is an n n matrix of nonnegative integers for which every row and column sums to r n = 3, r = 7 n = 3, r = 6
6 Magic Squares and Syzygies p. 4 A basic question How many n n magic squares have line sum r? Call this number H n (r).
7 Magic Squares and Syzygies p. 4 A basic question How many n n magic squares have line sum r? Call this number H n (r). Trivial case: H n (0) = 1.
8 Magic Squares and Syzygies p. 4 A basic question How many n n magic squares have line sum r? Call this number H n (r). Trivial case: H n (0) =
9 Magic Squares and Syzygies p. 5 The cases n = 1, n = 2 H 1 (r) = 1: [r]
10 Magic Squares and Syzygies p. 5 The cases n = 1, n = 2 H 1 (r) = 1: [r] What about n = 2? Let 0 i r. [ i??? ]
11 Magic Squares and Syzygies p. 5 The cases n = 1, n = 2 H 1 (r) = 1: [r] What about n = 2? Let 0 i r. [ ] i r i r i?
12 Magic Squares and Syzygies p. 5 The cases n = 1, n = 2 H 1 (r) = 1: [r] What about n = 2? Let 0 i r. [ ] i r i r i i
13 Magic Squares and Syzygies p. 5 The cases n = 1, n = 2 H 1 (r) = 1: [r] What about n = 2? Let 0 i r. [ i r i r i i Hence H 2 (r) = r + 1 (r + 1 choices for 0 i r). ]
14 Magic Squares and Syzygies p. 6 The case r = 1 If every row and column of an n n magic square M sums to 1, then M has one 1 in every row and column and 0 s elsewhere, i.e., M is a permutation matrix.
15 Magic Squares and Syzygies p. 6 The case r = 1 If every row and column of an n n magic square M sums to 1, then M has one 1 in every row and column and 0 s elsewhere, i.e., M is a permutation matrix. n choices for where to put 1 in the first row. Then n 1 choices for 1 in the second row. Then n 2 choices for 1 in the third row. Etc.
16 Magic Squares and Syzygies p. 6 The case r = 1 If every row and column of an n n magic square M sums to 1, then M has one 1 in every row and column and 0 s elsewhere, i.e., M is a permutation matrix. n choices for where to put 1 in the first row. Then n 1 choices for 1 in the second row. Then n 2 choices for 1 in the third row. Etc. Thus H n (1) = n(n 1)(n 2) 1 = n!.
17 Magic Squares and Syzygies p. 7 Binomial coefficients Let 0 k n. Define ( ) n n(n 1)(n 2) (n k + 1) =. k k!
18 Magic Squares and Syzygies p. 7 Binomial coefficients Let 0 k n. Define ( ) n n(n 1)(n 2) (n k + 1) =. k k! ( ) n n(n 1) Examples. = = (n2 n) ( ) n = (n + 2)(n + 1)n 6 = 1 6 (n3 + 3n 2 + 2n).
19 Magic Squares and Syzygies p. 7 Binomial coefficients Let 0 k n. Define ( ) n n(n 1)(n 2) (n k + 1) =. k k! ( ) n n(n 1) Examples. = = (n2 n) ( ) n + 2 (n + 2)(n + 1)n = = (n3 + 3n 2 + 2n). In general, ( n+j k ) is a polynomial in n, degree k.
20 Magic Squares and Syzygies p. 8 Let s get serious Theorem (Birkhoff-von Neumann, 1946, 1952). Every n n magic square with row and column sum r is a sum of r permutation matrices (of size n n).
21 Magic Squares and Syzygies p. 8 Let s get serious Theorem (Birkhoff-von Neumann, 1946, 1952). Every n n magic square with row and column sum r is a sum of r permutation matrices (of size n n) =
22 Magic Squares and Syzygies p : first approximation Every 3 3 magic square with row and column sum r is a sum of r 3 3 permutation matrices (six such matrices).
23 Magic Squares and Syzygies p : first approximation Every 3 3 magic square with row and column sum r is a sum of r 3 3 permutation matrices (six such matrices). If all these sums were different, then from Combinatorics 101 we would have ( ) r + 5 H 3 (r) =. 5
24 Magic Squares and Syzygies p : first approximation Every 3 3 magic square with row and column sum r is a sum of r 3 3 permutation matrices (six such matrices). If all these sums were different, then from Combinatorics 101 we would have ( ) r + 5 H 3 (r) =. 5 However, the sums are not all different!
25 Magic Squares and Syzygies p. 10 A relation = =
26 Magic Squares and Syzygies p. 11 Syzygies Such a relation is called a syzygy.
27 Magic Squares and Syzygies p. 11 Syzygies Such a relation is called a syzygy. from suzugos (σύζυγoζ), yoked together
28 Astronomical significance Magic Squares and Syzygies p. 12
29 Magic Squares and Syzygies p. 13 Correction term The relation (syzygy) between the six 3 3 permutation matrices means that our first approximation H 3 (r) = ( ) r+5 5 is an overcount. Get a second approximation ( ) ( ) r + 5 r + 2 H 3 (r) = 5 5 = 1 8 (r4 + 6r r r + 8) = 1 8 (r + 1)(r + 2)(r2 + 3r + 4).
30 Magic Squares and Syzygies p. 14 No further corrections! No further syzygies, so this is correct!
31 Magic Squares and Syzygies p. 14 No further corrections! No further syzygies, so this is correct! First proved by Percy Alexander MacMahon ( ) c. 1916, as part of his syzygetic method. First real result on H n (r).
32 Percy Alexander MacMahon Magic Squares and Syzygies p. 15
33 Magic Squares and Syzygies p. 16 Anand-Dumir-Gupta (1966) Conjecture (1966). For any n, H n (r) is a polynomial in r of degree (n 1) 2. Moreover, H n ( 1) = H n ( 2) = = H n ( n + 1) = 0 H n (r) = ( 1) n 1 H n ( n r).
34 Magic Squares and Syzygies p. 16 Anand-Dumir-Gupta (1966) Conjecture (1966). For any n, H n (r) is a polynomial in r of degree (n 1) 2. Moreover, H n ( 1) = H n ( 2) = = H n ( n + 1) = 0 H n (r) = ( 1) n 1 H n ( n r). Example. H 3 (r) = 1 8 (r + 1)(r + 2)(r2 + 3r + 4)
35 Magic Squares and Syzygies p What about 4 4 magic squares?
36 Magic Squares and Syzygies p What about 4 4 magic squares? There are permutation matrices, giving a first approximation ( ) r
37 Magic Squares and Syzygies p What about 4 4 magic squares? There are permutation matrices, giving a first approximation ( ) r Now, however, there are 178 independent syzygies with varying numbers of terms.
38 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies.
39 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies.
40 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies. Then 7416 third-order
41 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies. Then 7416 third-order, fourth order
42 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies. Then 7416 third-order, fourth order, 25144
43 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies. Then 7416 third-order, fourth order, 25144, 35562
44 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies. The 7416 third-order, fourth order, 25144, 35562, 42204
45 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies. Then 7416 third-order, fourth order, 25144, 35562, 42204, 35562
46 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies. Then 7416 third-order, fourth order, 25144, 35562, 42204, 35562, 25144
47 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies. Then 7416 third-order, fourth order, 25144, 35562, 42204, 35562, 25144, 16440
48 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies. Then 7416 third-order, fourth order, 25144, 35562, 42204, 35562, 25144, 16440, 7416
49 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies. Then 7416 third-order, fourth order, 25144, 35562, 42204, 35562, 25144, 16440, 7416, 1837
50 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies. Then 7416 third-order, fourth order, 25144, 35562, 42204, 35562, 25144, 16440, 7416, 1837, 178
51 Magic Squares and Syzygies p. 18 Not the end of the story These 178 syzygies have relations among them, called second-order syzygies. There are 1837 second-order syzygies. Then 7416 third-order, fourth order, 25144, 35562, 42204, 35562, 25144, 16440, 7416, 1837, 178, and finally ending in one 14th order syzygy!
52 Magic Squares and Syzygies p. 19 Generalities To define syzygies rigorously requires commutative algebra. David Hilbert ( ): proved famous Hilbert syzygy theorem (1890).
53 Magic Squares and Syzygies p. 19 Generalities To define syzygies rigorously requires commutative algebra. David Hilbert ( ): proved famous Hilbert syzygy theorem (1890). For n n magic squares, guarantees that the syzygy process will come to an end in at most n! steps.
54 Magic Squares and Syzygies p. 19 Generalities To define syzygies rigorously requires commutative algebra. David Hilbert ( ): proved famous Hilbert syzygy theorem (1890). For n n magic squares, guarantees that the syzygy process will come to an end in at most n! steps. This implies that H n (r) is indeed a polynomial in r, at least for r sufficiently large. Extra tweaking needed for all r.
55 Magic Squares and Syzygies p. 20 More refined results More sophisticated algebra (Cohen-Macaulay rings, Auslander-Buchsbaum theorem): the chain of syzygies ends in exactly n! n 2 + 2n 2 steps.
56 Magic Squares and Syzygies p. 20 More refined results More sophisticated algebra (Cohen-Macaulay rings, Auslander-Buchsbaum theorem): the chain of syzygies ends in exactly n! n 2 + 2n 2 steps. Number of syzygies at each step for n = 4: 1, 178, 1837, 7416, 16440, 25144, 35562, 42204, 35562, 25144, 16440, 7416, 1837, 178, 1
57 Magic Squares and Syzygies p. 20 More refined results More sophisticated algebra (Cohen-Macaulay rings, Auslander-Buchsbaum theorem): the chain of syzygies ends in exactly n! n 2 + 2n 2 steps. Number of syzygies at each step for n = 4: 1, 178, 1837, 7416, 16440, 25144, 35562, 42204, 35562, 25144, 16440, 7416, 1837, 178, 1 Note symmetry! (Sequence is palindromic.)
58 Magic Squares and Syzygies p. 21 Symmetry Key reason for symmetry: let J be the n n all 1 s matrix. Then M is an n n magic square with row and column sums r if and only if M + J is an n n magic square with positive entries and row and column sums r + n.
59 Magic Squares and Syzygies p. 21 Symmetry Key reason for symmetry: let J be the n n all 1 s matrix. Then M is an n n magic square with row and column sums r if and only if M + J is an n n magic square with positive entries and row and column sums r + n. Symmetry implies the remainder of the Anand-Dumir-Gupta conjecture: H n ( 1) = H n ( 2) = = H n ( n + 1) = 0 H n (r) = ( 1) n 1 H n ( n r).
60 5 5 Magic Squares and Syzygies p. 22
61 Magic Squares and Syzygies p over syzygies!
62 Magic Squares and Syzygies p over syzygies! Mankind may never know the exact number.
63 Magic Squares and Syzygies p. 23 Another approach Every 3 3 magic square has exactly one of the forms: a + e b + d c c + d a b + e b c + e a + d, a b + d c + e + 1 c + d a + e + 1 b b + e + 1 c a + d a + d + 1 b c + e + 1 c a + e + 1 b + d + 1 b + e + 1 c + d + 1 a,, where a,b,c,d,e 0.
64 Magic Squares and Syzygies p. 24 Further formulas This gives: H 3 (r) = ( ) r ( ) r ( ) r
65 Magic Squares and Syzygies p. 24 Further formulas This gives: H 3 (r) = ( ) r ( ) r ( ) r This type of argument can be done for any n, using a geometric approach (shellability).
66 Magic Squares and Syzygies p. 24 Further formulas This gives: H 3 (r) = ( ) r ( ) r ( ) r This type of argument can be done for any n, using a geometric approach (shellability). Regard an n n magic square as lying in the n 2 -dimensional space R n2.
67 Magic Squares and Syzygies p For 4 4 magic squares we get ( ) ( ) ( ) ( ) r + 9 r + 8 r + 7 r ( ) ( ) ( ) r + 5 r + 4 r
68 Magic Squares and Syzygies p For 4 4 magic squares we get ( ) ( ) ( ) ( ) r + 9 r + 8 r + 7 r ( ) ( ) ( ) r + 5 r + 4 r (352 cases in all)
69 Magic Squares and Syzygies p For 4 4 magic squares we get ( ) ( ) ( ) ( ) r + 9 r + 8 r + 7 r ( ) ( ) ( ) r + 5 r + 4 r (352 cases in all) For n = 5 there are cases!
70 Magic Squares and Syzygies p For 4 4 magic squares we get ( ) ( ) ( ) ( ) r + 9 r + 8 r + 7 r ( ) ( ) ( ) r + 5 r + 4 r (352 cases in all) For n = 5 there are cases! Note symmetry of (1, 14, 87, 148, 87, 14, 1). Equivalent to previous symmetry phenomena.
71 Magic Squares and Syzygies p. 26 Computation Symmetry property useful for computation.
72 Magic Squares and Syzygies p. 26 Computation Symmetry property useful for computation. Example. H 4 (r) is a polynomial of degree 9. Ten values are needed to compute H 4 (r), say H 4 (0) = 1,H 4 (1) = 24,H 4 (2),...,H 4 (9).
73 Magic Squares and Syzygies p. 26 Computation Symmetry property useful for computation. Example. H 4 (r) is a polynomial of degree 9. Ten values are needed to compute H 4 (r), say H 4 (0) = 1,H 4 (1) = 24,H 4 (2),...,H 4 (9). But H 4 ( 1) = H 4 ( 2) = H 4 ( 3) = 0, H 4 (0) = H 4 ( 4),H 4 (1) = H 4 ( 5),H 4 (2) = H 4 ( 6),H 4 (3) = H 4 ( 7). Thus only need to compute H 4 (0) = 1,H 4 (1) = 24,H 4 (2),H 4 (3).
74 Magic Squares and Syzygies p. 27 Unimodality (1, 14, 87, 148, 87, 14, 1) Note (unimodality).
75 Magic Squares and Syzygies p. 27 Unimodality (1, 14, 87, 148, 87, 14, 1) Note (unimodality). True for any n. Very deep proof: toric varieties, intersection homology, hard Lefschetz theorem,...
76 Magic Squares and Syzygies p. 28 Zeros of H n (r) H 9 (r) is a polynomial of degree 64 and thus has 64 complex zeros (or roots) z, i.e., H 9 (z) = 0.
77 Magic Squares and Syzygies p. 28 Zeros of H n (r) H 9 (r) is a polynomial of degree 64 and thus has 64 complex zeros (or roots) z, i.e., H 9 (z) = 0. Zeros of H_9(n)
78 Magic Squares and Syzygies p. 29 First variation Holey magic squares: some entries (or holes) are specified to be 0.
79 Magic Squares and Syzygies p. 29 First variation Holey magic squares: some entries (or holes) are specified to be 0. Example
80 Magic Squares and Syzygies p. 30 H n,s (r) H n,s (r): number of n n magic squares with line sum r and hole set S.
81 Magic Squares and Syzygies p. 30 H n,s (r) H n,s (r): number of n n magic squares with line sum r and hole set S. 0 Example. 0 0
82 Magic Squares and Syzygies p. 30 H n,s (r) H n,s (r): number of n n magic squares with line sum r and hole set S. 0 Example i r : so H 3,S (r) = r i r i r i 0 i i r i 0,
83 Magic Squares and Syzygies p. 31 Polynomiality If a magic square with holes is written as a sum of permutation matrices P, then each P still has these holes.
84 Magic Squares and Syzygies p. 31 Polynomiality If a magic square with holes is written as a sum of permutation matrices P, then each P still has these holes. Thus Birkhoff-von Neumann, syzygy, and shellability arguments still apply: Theorem. H n,s (r) is a polynomial in r.
85 Magic Squares and Syzygies p. 32 An example M =
86 Magic Squares and Syzygies p. 32 An example M = H 4,S (r) = 1 24 (r + 1)(r + 2)(r + 3)(r2 + 3r + 4) ( ) ( ) ( ) r + 5 r + 4 r + 3 =
87 Magic Squares and Syzygies p. 32 An example M = H 4,S (r) = 1 24 (r + 1)(r + 2)(r + 3)(r2 + 3r + 4) ( ) ( ) ( ) r + 5 r + 4 r + 3 = NOTE. The sequence (1, 2, 2) is not symmetric!
88 Magic Squares and Syzygies p. 33 Minimal positive magic squares ( ) r + 5 H 4,S (r) = 1 5 ( ) r ( ) r
89 Magic Squares and Syzygies p. 33 Minimal positive magic squares ( ) r + 5 H 4,S (r) = 1 5 ( ) r ( ) r There are two minimal positive magic squares:
90 Magic Squares and Syzygies p. 34 Reciprocity ( ) r + 5 H 4,S (r) = 1 5 ( ) r ( ) r
91 Magic Squares and Syzygies p. 34 Reciprocity ( ) r + 5 H 4,S (r) = 1 5 ( ) r ( ) r In fact: let H 4,S (r) be the number of positive (except for the holes S) 4 4 magic squares with hole set S and line sum r.
92 Magic Squares and Syzygies p. 34 Reciprocity ( ) r + 5 H 4,S (r) = 1 5 ( ) r ( ) r In fact: let H 4,S (r) be the number of positive (except for the holes S) 4 4 magic squares with hole set S and line sum r. ( ) ( ) ( ) r + 1 r r 1 H 4,S (r) =
93 Magic Squares and Syzygies p. 34 Reciprocity ( ) r + 5 H 4,S (r) = 1 5 ( ) r ( ) r In fact: let H 4,S (r) be the number of positive (except for the holes S) 4 4 magic squares with hole set S and line sum r. ( ) ( ) ( ) r + 1 r r 1 H 4,S (r) = (1, 2, 2) and (2, 2, 1) are reverses!
94 Magic Squares and Syzygies p. 35 Second variation S n (r): number of n n symmetric magic squares with line sums equal to r
95 Magic Squares and Syzygies p. 35 Second variation S n (r): number of n n symmetric magic squares with line sums equal to r
96 Magic Squares and Syzygies p. 35 Second variation S n (r): number of n n symmetric magic squares with line sums equal to r
97 Magic Squares and Syzygies p. 36 Not a polynomial! S 3 (r) = { 1 8 (2r3 + 9r r + 8), r even 1 8 (2r3 + 9r r + 7), r odd
98 Magic Squares and Syzygies p. 36 Not a polynomial! S 3 (r) = { 1 8 (2r3 + 9r r + 8), r even 1 8 (2r3 + 9r r + 7), r odd Theorem. For any n 1 there exist polynomials P n (r) and Q n (r) such that S n (r) = { Pn (r), r even Q n (r), r odd.
99 Magic Squares and Syzygies p. 37 Birkhoff-von Neumann fails False that every symmetric magic square M is a sum of symmetric permutation matrices: =? 1 1 0
100 Magic Squares and Syzygies p. 38 Why r even and r odd However 2M is sum of symmetric magic squares of row and column sum two: M = P P k (permutation matrices) 2M = M + M t = (P 1 + P t 1) + + (P k + P t k).
101 Magic Squares and Syzygies p. 39 Third variation I n (r): number of n n n magic cubes (rows, columns, and pillars sum to r)
102 Magic Squares and Syzygies p. 39 Third variation I n (r): number of n n n magic cubes (rows, columns, and pillars sum to r) No analogue of Birkhoff-von Neumann theorem. Very general results give: Theorem. For every n 1 there is a p (depending on n) and polynomials T n,0 (r),...,t n,p 1 (r) such that I n (r) = T n,i (r) if r i (mod p).
103 Magic Squares and Syzygies p. 39 Third variation I n (r): number of n n n magic cubes (rows, columns, and pillars sum to r) No analogue of Birkhoff-von Neumann theorem. Very general results give: Theorem. For every n 1 there is a p (depending on n) and polynomials T n,0 (r),...,t n,p 1 (r) such that I n (r) = T n,i (r) if r i (mod p). Value of p not known in general.
104 Magic Squares and Syzygies p : p = 2 I 3 (r) = ( 18r r r r r r r r ), r even
105 Magic Squares and Syzygies p : p = 2 I 3 (r) = ( 18r r r r r r r r ), r even I 3 (r) = ( r ), r odd
106 Magic Squares and Syzygies p : p = 2 I 3 (r) = ( 18r r r r r r r r ), r even I 3 (r) = : p =? ( r ), r odd
107 The last slide Magic Squares and Syzygies p. 41
108 The last slide Magic Squares and Syzygies p. 42
109 The last slide Magic Squares and Syzygies p. 42
Zeros of a Polynomial Function
Zeros of a Polynomial Function An important consequence of the Factor Theorem is that finding the zeros of a polynomial is really the same thing as factoring it into linear factors. In this section we
Solution to Homework 2
Solution to Homework 2 Olena Bormashenko September 23, 2011 Section 1.4: 1(a)(b)(i)(k), 4, 5, 14; Section 1.5: 1(a)(b)(c)(d)(e)(n), 2(a)(c), 13, 16, 17, 18, 27 Section 1.4 1. Compute the following, if
SECTION 10-2 Mathematical Induction
73 0 Sequences and Series 6. Approximate e 0. using the first five terms of the series. Compare this approximation with your calculator evaluation of e 0.. 6. Approximate e 0.5 using the first five terms
CHAPTER 3. Methods of Proofs. 1. Logical Arguments and Formal Proofs
CHAPTER 3 Methods of Proofs 1. Logical Arguments and Formal Proofs 1.1. Basic Terminology. An axiom is a statement that is given to be true. A rule of inference is a logical rule that is used to deduce
Math 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Fall 2011 Homework # 5, due Wednesday, February 22 5.1.4 Let P (n) be the statement that 1 3 + 2 3 + + n 3 = (n(n + 1)/2) 2 for the positive integer n. a) What
PYTHAGOREAN TRIPLES KEITH CONRAD
PYTHAGOREAN TRIPLES KEITH CONRAD 1. Introduction A Pythagorean triple is a triple of positive integers (a, b, c) where a + b = c. Examples include (3, 4, 5), (5, 1, 13), and (8, 15, 17). Below is an ancient
Linearly Independent Sets and Linearly Dependent Sets
These notes closely follow the presentation of the material given in David C. Lay s textbook Linear Algebra and its Applications (3rd edition). These notes are intended primarily for in-class presentation
Homework until Test #2
MATH31: Number Theory Homework until Test # Philipp BRAUN Section 3.1 page 43, 1. It has been conjectured that there are infinitely many primes of the form n. Exhibit five such primes. Solution. Five such
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009. Notes on Algebra
U.C. Berkeley CS276: Cryptography Handout 0.1 Luca Trevisan January, 2009 Notes on Algebra These notes contain as little theory as possible, and most results are stated without proof. Any introductory
k, then n = p2α 1 1 pα k
Powers of Integers An integer n is a perfect square if n = m for some integer m. Taking into account the prime factorization, if m = p α 1 1 pα k k, then n = pα 1 1 p α k k. That is, n is a perfect square
Chapter 17. Orthogonal Matrices and Symmetries of Space
Chapter 17. Orthogonal Matrices and Symmetries of Space Take a random matrix, say 1 3 A = 4 5 6, 7 8 9 and compare the lengths of e 1 and Ae 1. The vector e 1 has length 1, while Ae 1 = (1, 4, 7) has length
3. Mathematical Induction
3. MATHEMATICAL INDUCTION 83 3. Mathematical Induction 3.1. First Principle of Mathematical Induction. Let P (n) be a predicate with domain of discourse (over) the natural numbers N = {0, 1,,...}. If (1)
1.2. Successive Differences
1. An Application of Inductive Reasoning: Number Patterns In the previous section we introduced inductive reasoning, and we showed how it can be applied in predicting what comes next in a list of numbers
a 11 x 1 + a 12 x 2 + + a 1n x n = b 1 a 21 x 1 + a 22 x 2 + + a 2n x n = b 2.
Chapter 1 LINEAR EQUATIONS 1.1 Introduction to linear equations A linear equation in n unknowns x 1, x,, x n is an equation of the form a 1 x 1 + a x + + a n x n = b, where a 1, a,..., a n, b are given
Prime numbers and prime polynomials. Paul Pollack Dartmouth College
Prime numbers and prime polynomials Paul Pollack Dartmouth College May 1, 2008 Analogies everywhere! Analogies in elementary number theory (continued fractions, quadratic reciprocity, Fermat s last theorem)
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS. + + x 2. x n. a 11 a 12 a 1n b 1 a 21 a 22 a 2n b 2 a 31 a 32 a 3n b 3. a m1 a m2 a mn b m
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS 1. SYSTEMS OF EQUATIONS AND MATRICES 1.1. Representation of a linear system. The general system of m equations in n unknowns can be written a 11 x 1 + a 12 x 2 +
Similarity and Diagonalization. Similar Matrices
MATH022 Linear Algebra Brief lecture notes 48 Similarity and Diagonalization Similar Matrices Let A and B be n n matrices. We say that A is similar to B if there is an invertible n n matrix P such that
Row Echelon Form and Reduced Row Echelon Form
These notes closely follow the presentation of the material given in David C Lay s textbook Linear Algebra and its Applications (3rd edition) These notes are intended primarily for in-class presentation
Lecture 13 - Basic Number Theory.
Lecture 13 - Basic Number Theory. Boaz Barak March 22, 2010 Divisibility and primes Unless mentioned otherwise throughout this lecture all numbers are non-negative integers. We say that A divides B, denoted
1 Solving LPs: The Simplex Algorithm of George Dantzig
Solving LPs: The Simplex Algorithm of George Dantzig. Simplex Pivoting: Dictionary Format We illustrate a general solution procedure, called the simplex algorithm, by implementing it on a very simple example.
SOLVING POLYNOMIAL EQUATIONS BY RADICALS
SOLVING POLYNOMIAL EQUATIONS BY RADICALS Lee Si Ying 1 and Zhang De-Qi 2 1 Raffles Girls School (Secondary), 20 Anderson Road, Singapore 259978 2 Department of Mathematics, National University of Singapore,
= 2 + 1 2 2 = 3 4, Now assume that P (k) is true for some fixed k 2. This means that
Instructions. Answer each of the questions on your own paper, and be sure to show your work so that partial credit can be adequately assessed. Credit will not be given for answers (even correct ones) without
Math Common Core Sampler Test
High School Algebra Core Curriculum Math Test Math Common Core Sampler Test Our High School Algebra sampler covers the twenty most common questions that we see targeted for this level. For complete tests
Algebra 2 Chapter 1 Vocabulary. identity - A statement that equates two equivalent expressions.
Chapter 1 Vocabulary identity - A statement that equates two equivalent expressions. verbal model- A word equation that represents a real-life problem. algebraic expression - An expression with variables.
Just the Factors, Ma am
1 Introduction Just the Factors, Ma am The purpose of this note is to find and study a method for determining and counting all the positive integer divisors of a positive integer Let N be a given positive
Mathematical Induction
Mathematical Induction (Handout March 8, 01) The Principle of Mathematical Induction provides a means to prove infinitely many statements all at once The principle is logical rather than strictly mathematical,
Chapter 3. Distribution Problems. 3.1 The idea of a distribution. 3.1.1 The twenty-fold way
Chapter 3 Distribution Problems 3.1 The idea of a distribution Many of the problems we solved in Chapter 1 may be thought of as problems of distributing objects (such as pieces of fruit or ping-pong balls)
CONTINUED FRACTIONS AND PELL S EQUATION. Contents 1. Continued Fractions 1 2. Solution to Pell s Equation 9 References 12
CONTINUED FRACTIONS AND PELL S EQUATION SEUNG HYUN YANG Abstract. In this REU paper, I will use some important characteristics of continued fractions to give the complete set of solutions to Pell s equation.
Math 1050 Khan Academy Extra Credit Algebra Assignment
Math 1050 Khan Academy Extra Credit Algebra Assignment KhanAcademy.org offers over 2,700 instructional videos, including hundreds of videos teaching algebra concepts, and corresponding problem sets. In
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS
MATRIX ALGEBRA AND SYSTEMS OF EQUATIONS Systems of Equations and Matrices Representation of a linear system The general system of m equations in n unknowns can be written a x + a 2 x 2 + + a n x n b a
Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013
Module MA3411: Abstract Algebra Galois Theory Appendix Michaelmas Term 2013 D. R. Wilkins Copyright c David R. Wilkins 1997 2013 Contents A Cyclotomic Polynomials 79 A.1 Minimum Polynomials of Roots of
8 Primes and Modular Arithmetic
8 Primes and Modular Arithmetic 8.1 Primes and Factors Over two millennia ago already, people all over the world were considering the properties of numbers. One of the simplest concepts is prime numbers.
Matrix Algebra. Some Basic Matrix Laws. Before reading the text or the following notes glance at the following list of basic matrix algebra laws.
Matrix Algebra A. Doerr Before reading the text or the following notes glance at the following list of basic matrix algebra laws. Some Basic Matrix Laws Assume the orders of the matrices are such that
BookTOC.txt. 1. Functions, Graphs, and Models. Algebra Toolbox. Sets. The Real Numbers. Inequalities and Intervals on the Real Number Line
College Algebra in Context with Applications for the Managerial, Life, and Social Sciences, 3rd Edition Ronald J. Harshbarger, University of South Carolina - Beaufort Lisa S. Yocco, Georgia Southern University
Continued Fractions and the Euclidean Algorithm
Continued Fractions and the Euclidean Algorithm Lecture notes prepared for MATH 326, Spring 997 Department of Mathematics and Statistics University at Albany William F Hammond Table of Contents Introduction
Orthogonal Diagonalization of Symmetric Matrices
MATH10212 Linear Algebra Brief lecture notes 57 Gram Schmidt Process enables us to find an orthogonal basis of a subspace. Let u 1,..., u k be a basis of a subspace V of R n. We begin the process of finding
2 When is a 2-Digit Number the Sum of the Squares of its Digits?
When Does a Number Equal the Sum of the Squares or Cubes of its Digits? An Exposition and a Call for a More elegant Proof 1 Introduction We will look at theorems of the following form: by William Gasarch
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus
South Carolina College- and Career-Ready (SCCCR) Pre-Calculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
Sample Induction Proofs
Math 3 Worksheet: Induction Proofs III, Sample Proofs A.J. Hildebrand Sample Induction Proofs Below are model solutions to some of the practice problems on the induction worksheets. The solutions given
Algebra 1 Course Title
Algebra 1 Course Title Course- wide 1. What patterns and methods are being used? Course- wide 1. Students will be adept at solving and graphing linear and quadratic equations 2. Students will be adept
Solutions for Practice problems on proofs
Solutions for Practice problems on proofs Definition: (even) An integer n Z is even if and only if n = 2m for some number m Z. Definition: (odd) An integer n Z is odd if and only if n = 2m + 1 for some
As we have seen, there is a close connection between Legendre symbols of the form
Gauss Sums As we have seen, there is a close connection between Legendre symbols of the form 3 and cube roots of unity. Secifically, if is a rimitive cube root of unity, then 2 ± i 3 and hence 2 2 3 In
26. Determinants I. 1. Prehistory
26. Determinants I 26.1 Prehistory 26.2 Definitions 26.3 Uniqueness and other properties 26.4 Existence Both as a careful review of a more pedestrian viewpoint, and as a transition to a coordinate-independent
n 2 + 4n + 3. The answer in decimal form (for the Blitz): 0, 75. Solution. (n + 1)(n + 3) = n + 3 2 lim m 2 1
. Calculate the sum of the series Answer: 3 4. n 2 + 4n + 3. The answer in decimal form (for the Blitz):, 75. Solution. n 2 + 4n + 3 = (n + )(n + 3) = (n + 3) (n + ) = 2 (n + )(n + 3) ( 2 n + ) = m ( n
PUTNAM TRAINING POLYNOMIALS. Exercises 1. Find a polynomial with integral coefficients whose zeros include 2 + 5.
PUTNAM TRAINING POLYNOMIALS (Last updated: November 17, 2015) Remark. This is a list of exercises on polynomials. Miguel A. Lerma Exercises 1. Find a polynomial with integral coefficients whose zeros include
arxiv:0909.4913v2 [math.ho] 4 Nov 2009
![arxiv:0909.4913v2 [math.ho] 4 Nov 2009](/thumbs/40/21097349.jpg "arxiv:0909.4913v2 [math.ho] 4 Nov 2009")
IRRATIONALITY FROM THE BOOK STEVEN J. MILLER AND DAVID MONTAGUE arxiv:0909.4913v2 [math.ho] 4 Nov 2009 A right of passage to theoretical mathematics is often a proof of the irrationality of 2, or at least
Trigonometric Functions and Equations
Contents Trigonometric Functions and Equations Lesson 1 Reasoning with Trigonometric Functions Investigations 1 Proving Trigonometric Identities... 271 2 Sum and Difference Identities... 276 3 Extending
9.2 Summation Notation
9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a
Zeros of Polynomial Functions
Zeros of Polynomial Functions The Rational Zero Theorem If f (x) = a n x n + a n-1 x n-1 + + a 1 x + a 0 has integer coefficients and p/q (where p/q is reduced) is a rational zero, then p is a factor of
Refined enumerations of alternating sign matrices. Ilse Fischer. Universität Wien
Refined enumerations of alternating sign matrices Ilse Fischer Universität Wien 1 Central question Which enumeration problems have a solution in terms of a closed formula that (for instance) only involves
Notes on Orthogonal and Symmetric Matrices MENU, Winter 2013
Notes on Orthogonal and Symmetric Matrices MENU, Winter 201 These notes summarize the main properties and uses of orthogonal and symmetric matrices. We covered quite a bit of material regarding these topics,
THE SIGN OF A PERMUTATION
THE SIGN OF A PERMUTATION KEITH CONRAD 1. Introduction Throughout this discussion, n 2. Any cycle in S n is a product of transpositions: the identity (1) is (12)(12), and a k-cycle with k 2 can be written
2x + y = 3. Since the second equation is precisely the same as the first equation, it is enough to find x and y satisfying the system
1. Systems of linear equations We are interested in the solutions to systems of linear equations. A linear equation is of the form 3x 5y + 2z + w = 3. The key thing is that we don t multiply the variables
Algebra II End of Course Exam Answer Key Segment I. Scientific Calculator Only
Algebra II End of Course Exam Answer Key Segment I Scientific Calculator Only Question 1 Reporting Category: Algebraic Concepts & Procedures Common Core Standard: A-APR.3: Identify zeros of polynomials
SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH
31 Kragujevac J. Math. 25 (2003) 31 49. SHARP BOUNDS FOR THE SUM OF THE SQUARES OF THE DEGREES OF A GRAPH Kinkar Ch. Das Department of Mathematics, Indian Institute of Technology, Kharagpur 721302, W.B.,
This unit will lay the groundwork for later units where the students will extend this knowledge to quadratic and exponential functions.
Algebra I Overview View unit yearlong overview here Many of the concepts presented in Algebra I are progressions of concepts that were introduced in grades 6 through 8. The content presented in this course
ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section
ALGEBRA 2 CRA 2 REVIEW - Chapters 1-6 Answer Section MULTIPLE CHOICE 1. ANS: C 2. ANS: A 3. ANS: A OBJ: 5-3.1 Using Vertex Form SHORT ANSWER 4. ANS: (x + 6)(x 2 6x + 36) OBJ: 6-4.2 Solving Equations by
Name: Section Registered In:
Name: Section Registered In: Math 125 Exam 3 Version 1 April 24, 2006 60 total points possible 1. (5pts) Use Cramer s Rule to solve 3x + 4y = 30 x 2y = 8. Be sure to show enough detail that shows you are
H/wk 13, Solutions to selected problems
H/wk 13, Solutions to selected problems Ch. 4.1, Problem 5 (a) Find the number of roots of x x in Z 4, Z Z, any integral domain, Z 6. (b) Find a commutative ring in which x x has infinitely many roots.
Factoring Polynomials
Factoring Polynomials Factoring Factoring is the process of writing a polynomial as the product of two or more polynomials. The factors of 6x 2 x 2 are 2x + 1 and 3x 2. In this section, we will be factoring
A characterization of trace zero symmetric nonnegative 5x5 matrices
A characterization of trace zero symmetric nonnegative 5x5 matrices Oren Spector June 1, 009 Abstract The problem of determining necessary and sufficient conditions for a set of real numbers to be the
Math 55: Discrete Mathematics
Math 55: Discrete Mathematics UC Berkeley, Spring 2012 Homework # 9, due Wednesday, April 11 8.1.5 How many ways are there to pay a bill of 17 pesos using a currency with coins of values of 1 peso, 2 pesos,
Zero: If P is a polynomial and if c is a number such that P (c) = 0 then c is a zero of P.
MATH 11011 FINDING REAL ZEROS KSU OF A POLYNOMIAL Definitions: Polynomial: is a function of the form P (x) = a n x n + a n 1 x n 1 + + a x + a 1 x + a 0. The numbers a n, a n 1,..., a 1, a 0 are called
DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH
DETERMINANTS IN THE KRONECKER PRODUCT OF MATRICES: THE INCIDENCE MATRIX OF A COMPLETE GRAPH CHRISTOPHER RH HANUSA AND THOMAS ZASLAVSKY Abstract We investigate the least common multiple of all subdeterminants,
Algebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 2012-13 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur
Probability and Statistics Prof. Dr. Somesh Kumar Department of Mathematics Indian Institute of Technology, Kharagpur Module No. #01 Lecture No. #15 Special Distributions-VI Today, I am going to introduce
Au = = = 3u. Aw = = = 2w. so the action of A on u and w is very easy to picture: it simply amounts to a stretching by 3 and 2, respectively.
Chapter 7 Eigenvalues and Eigenvectors In this last chapter of our exploration of Linear Algebra we will revisit eigenvalues and eigenvectors of matrices, concepts that were already introduced in Geometry
The last three chapters introduced three major proof techniques: direct,
CHAPTER 7 Proving Non-Conditional Statements The last three chapters introduced three major proof techniques: direct, contrapositive and contradiction. These three techniques are used to prove statements
by the matrix A results in a vector which is a reflection of the given
Eigenvalues & Eigenvectors Example Suppose Then So, geometrically, multiplying a vector in by the matrix A results in a vector which is a reflection of the given vector about the y-axis We observe that
The Determinant: a Means to Calculate Volume
The Determinant: a Means to Calculate Volume Bo Peng August 20, 2007 Abstract This paper gives a definition of the determinant and lists many of its well-known properties Volumes of parallelepipeds are
Linear Algebra I. Ronald van Luijk, 2012
Linear Algebra I Ronald van Luijk, 2012 With many parts from Linear Algebra I by Michael Stoll, 2007 Contents 1. Vector spaces 3 1.1. Examples 3 1.2. Fields 4 1.3. The field of complex numbers. 6 1.4.
1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]
![1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]](/thumbs/40/21231772.jpg "1 Homework 1. [p 0 q i+j +... + p i 1 q j+1 ] + [p i q j ] + [p i+1 q j 1 +... + p i+j q 0 ]")
1 Homework 1 (1) Prove the ideal (3,x) is a maximal ideal in Z[x]. SOLUTION: Suppose we expand this ideal by including another generator polynomial, P / (3, x). Write P = n + x Q with n an integer not
3.1. RATIONAL EXPRESSIONS
3.1. RATIONAL EXPRESSIONS RATIONAL NUMBERS In previous courses you have learned how to operate (do addition, subtraction, multiplication, and division) on rational numbers (fractions). Rational numbers
AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC
AN ALGORITHM FOR DETERMINING WHETHER A GIVEN BINARY MATROID IS GRAPHIC W. T. TUTTE. Introduction. In a recent series of papers [l-4] on graphs and matroids I used definitions equivalent to the following.
KEANSBURG SCHOOL DISTRICT KEANSBURG HIGH SCHOOL Mathematics Department. HSPA 10 Curriculum. September 2007
KEANSBURG HIGH SCHOOL Mathematics Department HSPA 10 Curriculum September 2007 Written by: Karen Egan Mathematics Supervisor: Ann Gagliardi 7 days Sample and Display Data (Chapter 1 pp. 4-47) Surveys and
Mathematical Induction. Mary Barnes Sue Gordon
Mathematics Learning Centre Mathematical Induction Mary Barnes Sue Gordon c 1987 University of Sydney Contents 1 Mathematical Induction 1 1.1 Why do we need proof by induction?.... 1 1. What is proof by
5.1 Radical Notation and Rational Exponents
Section 5.1 Radical Notation and Rational Exponents 1 5.1 Radical Notation and Rational Exponents We now review how exponents can be used to describe not only powers (such as 5 2 and 2 3 ), but also roots
Solving Systems of Linear Equations
LECTURE 5 Solving Systems of Linear Equations Recall that we introduced the notion of matrices as a way of standardizing the expression of systems of linear equations In today s lecture I shall show how
MATH10040 Chapter 2: Prime and relatively prime numbers
MATH10040 Chapter 2: Prime and relatively prime numbers Recall the basic definition: 1. Prime numbers Definition 1.1. Recall that a positive integer is said to be prime if it has precisely two positive
Basics of Polynomial Theory
3 Basics of Polynomial Theory 3.1 Polynomial Equations In geodesy and geoinformatics, most observations are related to unknowns parameters through equations of algebraic (polynomial) type. In cases where
Mathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
THE DIMENSION OF A VECTOR SPACE
THE DIMENSION OF A VECTOR SPACE KEITH CONRAD This handout is a supplementary discussion leading up to the definition of dimension and some of its basic properties. Let V be a vector space over a field
4.5 Linear Dependence and Linear Independence
4.5 Linear Dependence and Linear Independence 267 32. {v 1, v 2 }, where v 1, v 2 are collinear vectors in R 3. 33. Prove that if S and S are subsets of a vector space V such that S is a subset of S, then
Undergraduate Notes in Mathematics. Arkansas Tech University Department of Mathematics
Undergraduate Notes in Mathematics Arkansas Tech University Department of Mathematics An Introductory Single Variable Real Analysis: A Learning Approach through Problem Solving Marcel B. Finan c All Rights
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS
NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document
Every Positive Integer is the Sum of Four Squares! (and other exciting problems)
Every Positive Integer is the Sum of Four Squares! (and other exciting problems) Sophex University of Texas at Austin October 18th, 00 Matilde N. Lalín 1. Lagrange s Theorem Theorem 1 Every positive integer
Playing with Numbers
PLAYING WITH NUMBERS 249 Playing with Numbers CHAPTER 16 16.1 Introduction You have studied various types of numbers such as natural numbers, whole numbers, integers and rational numbers. You have also
Notes on Determinant
ENGG2012B Advanced Engineering Mathematics Notes on Determinant Lecturer: Kenneth Shum Lecture 9-18/02/2013 The determinant of a system of linear equations determines whether the solution is unique, without
MATH10212 Linear Algebra. Systems of Linear Equations. Definition. An n-dimensional vector is a row or a column of n numbers (or letters): a 1.
MATH10212 Linear Algebra Textbook: D. Poole, Linear Algebra: A Modern Introduction. Thompson, 2006. ISBN 0-534-40596-7. Systems of Linear Equations Definition. An n-dimensional vector is a row or a column
a 1 x + a 0 =0. (3) ax 2 + bx + c =0. (4)
ROOTS OF POLYNOMIAL EQUATIONS In this unit we discuss polynomial equations. A polynomial in x of degree n, where n 0 is an integer, is an expression of the form P n (x) =a n x n + a n 1 x n 1 + + a 1 x
1 Determinants and the Solvability of Linear Systems
1 Determinants and the Solvability of Linear Systems In the last section we learned how to use Gaussian elimination to solve linear systems of n equations in n unknowns The section completely side-stepped
Collinear Points in Permutations
Collinear Points in Permutations Joshua N. Cooper Courant Institute of Mathematics New York University, New York, NY József Solymosi Department of Mathematics University of British Columbia, Vancouver,
2.5 Zeros of a Polynomial Functions
.5 Zeros of a Polynomial Functions Section.5 Notes Page 1 The first rule we will talk about is Descartes Rule of Signs, which can be used to determine the possible times a graph crosses the x-axis and
Integer roots of quadratic and cubic polynomials with integer coefficients
Integer roots of quadratic and cubic polynomials with integer coefficients Konstantine Zelator Mathematics, Computer Science and Statistics 212 Ben Franklin Hall Bloomsburg University 400 East Second Street
Zachary Monaco Georgia College Olympic Coloring: Go For The Gold
Zachary Monaco Georgia College Olympic Coloring: Go For The Gold Coloring the vertices or edges of a graph leads to a variety of interesting applications in graph theory These applications include various
ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS
ON GALOIS REALIZATIONS OF THE 2-COVERABLE SYMMETRIC AND ALTERNATING GROUPS DANIEL RABAYEV AND JACK SONN Abstract. Let f(x) be a monic polynomial in Z[x] with no rational roots but with roots in Q p for
Theorem3.1.1 Thedivisionalgorithm;theorem2.2.1insection2.2 If m, n Z and n is a positive
Chapter 3 Number Theory 159 3.1 Prime Numbers Prime numbers serve as the basic building blocs in the multiplicative structure of the integers. As you may recall, an integer n greater than one is prime
How To Prove The Dirichlet Unit Theorem
Chapter 6 The Dirichlet Unit Theorem As usual, we will be working in the ring B of algebraic integers of a number field L. Two factorizations of an element of B are regarded as essentially the same if
T ( a i x i ) = a i T (x i ).
Chapter 2 Defn 1. (p. 65) Let V and W be vector spaces (over F ). We call a function T : V W a linear transformation form V to W if, for all x, y V and c F, we have (a) T (x + y) = T (x) + T (y) and (b)
Basics of Counting. The product rule. Product rule example. 22C:19, Chapter 6 Hantao Zhang. Sample question. Total is 18 * 325 = 5850
Basics of Counting 22C:19, Chapter 6 Hantao Zhang 1 The product rule Also called the multiplication rule If there are n 1 ways to do task 1, and n 2 ways to do task 2 Then there are n 1 n 2 ways to do