Part V, Abstract Algebra CS131 Mathematics for Computer Scientists II Note 29 RINGS AND FIELDS

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1 CS131 Part V, Abstract Algebra CS131 Mathematics for Computer Scientists II Note 29 RINGS AND FIELDS We now look at some algebraic structures which have more than one binary operation. Rings and fields have addition, subtraction and multiplication operations. In a field, division is also possible. Rings A ring consists of a set R with two binary operations + and. (the. is usually omitted when writing products) such that: (1) R is an abelian group under + (its identity element is usually denoted by 0 and called the zero element of R) (2) R is closed under multiplication (3) the operation. is associative (4) the following distributive laws hold for any a, b, c R: a(b + c) = ab + ac (b + c)a = ba + ca (5) R may or may not have an identity element under. and may or may not have inverse elements under. An identity under. in a ring R is an element 1 R with 1 0 and 1a = a = a1 for all a R. If R is a ring with an identity 1 under., then we say that an element a 1 of R is an inverse of a R if aa 1 = 1 = a 1 a A ring R is called commutative if ab = ba for all a, b R Examples of Rings: Each of the sets Z, R and C is a commutative ring with identity under the usual addition and multiplication operations. Let n 2 be an integer. The set Z n = {0, 1,..., n 1} of integers mod n is a commutative ring with identity under addition and multiplication mod n. In this ring we have ab = p and a + b = s 29 1

2 (1) (2) if and only if p and q are the unique integers with 0 p, q n 1 and ab p mod n, a + b q mod n. Let n 2 be an integer. The set of n n matrices with entries from R is a ring with identity under matrix addition and multiplication. The identity is the n n identity matrix and the zero element is the n n zero matrix. This ring is not commutative. The set of all even integers forms a commutative ring under the usual addition and multiplication of integers. This ring does not have an identity. A Boolean algebra becomes a ring when addition and multiplication are defined by: a + b = (a b ) (a b) a.b = a b Properties of Rings. For any elements a and b of a ring: (1) 0a = 0 = a0 (2) ( a)b = ab = a( b) (3) ( a)( b) = ab Examples of Inverses. In the ring Z 3 = {0, 1, 2} we have 1.1 = 1 and 2.2 = 1 so every nonzero element is its own inverse. In the ring Z 4 = {0, 1, 2, 3} the element 2 has no inverse. To see this we can check all the candidates: 0.2 = 0, 1.2 = 2, 2.2 = 0, 3.2 = 2. The Cayley table for multiplication in Z 6 is:

3 and we see from this that 1 and 5 are the only elements of this ring which have inverses. Fields A field F is a commutative ring F with identity in which every nonzero element has an inverse. Examples of Fields: the rational numbers Q with the usual addition and multiplication the real numbers R with the usual addition and multiplication the complex numbers C with the usual addition and multiplication the set Z p of integers mod p where p is a prime number is a field under addition and multiplication mod p Properties of Fields. (1) In a field, a product of two nonzero elements is nonzero, or equivalently ab = 0 a = 0 or b = 0 (2) If a, b, c are elements of a field and a 0, then the following cancellation law holds: ab = ac b = c Proof. (1) Suppose ab = 0. If a 0, then a has an inverse a 1 and so ab = 0 a 1 ab = a 1 0 1b = 0 b = 0. (2) If a 0, then: ab = ac a 1 ab = a 1 ac 1b = 1c b = c. Note that properties (1) and (2) of fields need not hold in general for rings. For example in the ring Z 4 we have 2.2 = 0 so a product of two nonzero elements can be zero. Similarly the cancellation law does not hold in Z 4 since 2.2 = 2.0 but 2 0. Proposition. prime. The integer ring Z n mod n is a field if and only if n is Proof. First suppose that p is prime. To show that Z p is a field we let m {0, 1,..., p 1} and suppose that m has no inverse in Z p. Then none of the p numbers 0m, 1m, 2m,..., (p 1)m 29 3

4 can be equal to 1 so this list must contain two numbers which are equal in Z p. Hence we have im jm (mod p) or (i j )m 0 (mod p) for some i, j with 0 < i j < p. Since p is prime one of the numbers i j or m must be a multiple of p and considering their ranges the only possibility is m = 0 Hence 0 is the only element with no inverse and so Z p is a field. To complete the proof we show that if n is not prime, then Z n is not a field. If n 2 is not prime then we can write n = qr for some q, r 2. But now we have two nonzero elements q and r whose product is the zero element of Z n. Since this is not possible in a field it follows that Z n is not a field. Other examples of finite fields can be constructed by taking a polynomial which has no root in Z p and adjoining a root. For example the polynomial x has no root in Z 3 so if we add one and call it i, say, then the set {a + ib a, b Z 3 } becomes a field if we define addition and multiplication by (a + ib) + (c + id) = (a + c) + i(b + d) (a + ib)(c + id) = (ac bd) + i(bc + ad) i.e. we write i 2 as 1. This gives an example of a field with 9 elements. Finite fields have applications to generating sequences of pseudo-random numbers. See D. Knuth, The Art of Computer Programming, Volume 2 (page 28) for details. ABSTRACT Content definition of ring, definition of field, properties of rings and fields In this Note, we study algebraic structures which are more complicated than groups. Rings and Fields have two binary operations compared with only one for a group. These structures were created in order to describe, among others, the properties of number systems. History Trygve Nagell was a Norwegian mathematician who published papers and books, in Swedish and English, on number theory. From the late 1920s onwards his chief interest was the study of algebraic numbers, and his 1931 study of algebraic rings was perhaps his most important contribution to abstract algebra. Emil Artin [ ] was an Austrian mathematician who made important contributions to the development of class field theory and the theory of hypercomplex numbers. He was one of the creators of modern abstract algebra. 29 4

5 Joseph Henry Wedderburn [ ] was a Scottish mathematician who opened new lines of thought in the subject of mathematical Fields and who had a deep influence on the development of modern algebra. 29 5