Math 514 homework #1 Due date: 19950830 D. J. Bernstein 19950823 Let R be a set with elements 0 and 1, a unary operation −, and binary operations + and ·. Recall that R is a ring if R satisfies every identity satisfied by Z, the set of integers. To prove that R is a ring it suffices to check just eight identities: 0+x = x, 1x = x, x+y = y +x, xy = yx, x+(y +z) = (x+y)+z, x(yz) = (xy)z, x + (−x) = 0, and x(y + z) = xy + xz. For example, C, the set of complex numbers, is a ring. Let x and z be elements of R. We say that x divides z in R if z = xy for some y ∈ R. Other phrases meaning the same thing: z is a multiple of x in R; x is a divisor of z in R. An element x ∈ R is a unit in R if xy = 1 for some y ∈ R. In other words, a unit is a divisor of 1. Since 0y = 0, it is not possible for 0 to be a unit unless 0 = 1. We call R a field if (1) all nonzero elements are units and (2) 0 ≠ 1. For example, Q, the set of rational numbers, is a field, as is C. But Z is not, since 2 is not a unit in Z. A nonzero element x ∈ R is a zero divisor in R if xy = 0 for some nonzero y ∈ R. Beware: everything is a divisor of zero, but most elements of R are not zero divisors. For example, 0 and 1 are not zero divisors. We call R a domain if (1) it has no zero divisors and (2) 0 ≠ 1. For example, any field is a domain. If 0 = 1, we call R a zero ring. If 0 = 1 then y = 1y = 0y = 0 for all y ∈ R, so R = {0}. A zero ring is neither a domain nor a field. Let S be a subset of R. We say that S is a subring of R if S has 0 and 1 and is closed under addition, negation, and multiplication. Then S is a ring under the same operations. We say that S is a subfield of R if (1) S is a subring of R and (2) S is a field. Exercise 1. Prove that the set \{a + b\sqrt{−19} : a ∈ Q, b ∈ Q\} is a subfield of C. If A and B are rings then A × B = {(a, b) : a ∈ A, b ∈ B} is a ring under componentwise operations: the componentwise product of (a, b) and (c, d), for example, is (ac, bd). Exercise 2. Consider the ring R = Z × Q. What are the units in R? What are the zero divisors in R? Exercise 3. Let E be any subset of C. Let K be the intersection of all subfields of C that contain E. Prove that K is a subfield of C. (Hence K is the smallest subfield of C containing E.)