# Ring theory

1. If E, F are fields and F is a subring of E, show each q in Aut(E/F) permutes the roots in E of each nonzero p(x) in F[x]. Hint if p(x)=a0+a1x+a2x^2+. . . +anx^n then p(x) has at most n roots in E. show that for z in E, p(z)=0 implies p(q(z))=0

2 If R is a commutative ring of prime characteristic p, show the function f:R-->R, f(r)=r^p is a ring homomorphism

3. If a,b are positive integers with greatest common divisor d and least common multiple m, show aZ+bZ=dZ and aZ intersect bZ=mZ

4. For the cyclic group (Z,+) = <1> prove the ideal of Z is principal

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#### Solution Preview

1. Suppose, q is an automorphism of E/F, that is, q(f)=f for all f in F. Let p(x) be a polynomial in F[x], and z be an element of E. Then, p(z) is an element of E, and we can consider q(p(z)). Let p(x)=a0+a1x+a2x^2+. . . +anx^n. Then

q(p(z))=q(a0)+q(a1z)+q(a2z^2)+...+q(anz^n)=a0+a1q(z)+a2[q(z)]^2+...+an[q(z)]^n

Suppose further that p(z)=0. Then, we have

q(p(z))a0+a1q(z)+a2[q(z)]^2+...+an[q(z)]^n=q(0)=0

and so q(z) is again a root ot p(x). Since q is an automirphism, it must be one-to-one, and since p(x) has at most n roots, by the pigeonhole principle, q just permutes the roots of p(x).

2. In a ring of prime characteristic p we have ...

#### Solution Summary

This posting exemplifies Ring theory.

Group Theory : Homomorphisms, Kernels, Isomorphisms and Fields

(a) If G1 and G2 are groups, define what it means to describe a function

h:G1 -> G2 as a homomorphism.

(b) If h: G1 ?> G2 is a homomorphism, define the kernel of h.Prove that the range of h is a subgroup of G2 , and that the kernel of h is a normal subgroup of G1.

(c) Let G be the group of 2x2 real matrices under addition. Prove that precisely two of the three functions below is a homomorphism. Identify the kernel and range of each of those that is a homomorphism.

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(d)For the two homomorphisms in (c), prove that the two quotient groups

G/N, where N is the corresponding kernel, are isomorphic.

6 (a) Let S ={...}. Prove that, under the usual operations of addition and multiplication, S is a field.

b) Let T={a+ b sqrt 3| a,b E Z}.

List the elements of T.

Addition and multiplication are defined on T in the natural way, with the coefficients being reduced modulo 3, as necessary. Prove that T is a commutative ring with a one.

By considering the element sqrt 3 prove that T is not a field.

Is T an integral domain?