abstract algebra: show num. of elts in finite gp G such that

[friday616]

I could definitely use some help with these algebra problems from my practice test. Any help would be appreciated. Thanks!

1. Let G be a finite group. Show that the number of elements, x, in G such that x^3=1 is odd.

2. Show that there are no finite fields with 10 elements.

3. Let f(x) be a monic polynomial over Z. Show that if n is a nonzero integer such that f(n) = 0, then n I f(0).

4. Show that if f and g are homomorphisms from Q into a ring R, Q is the field of rational numbers, then if f(m) = g(m) for every integer m, f = g.

5. Suppose R is a factorial domain and a and b are in R with a not zero and a and b relatively prime. Show that if a I bc, c in R, then a I c.

 

[daon]

1) Assume x is not 1.

\(\displaystyle x^3=1 \Rightarrow (x^{-1})^3=1\)

Each element and its inverse (which is unique) forms a pair.

Add in the identity (1, e, whatever) and it creates an odd number.

2) Assume F is a field containing 10 elements. F cannot contain zero divisors. Think about that...

3) Let f(x) = x^m + ... + ax + b
f(n) = n^m + ... + an + b = 0.
f(0) = b.

So we have n^m + ... + an + f(0) = 0.

I think you can finish.

4) Homomorphisms preserve additive and (if they exist) multiplicative inverses.

\(\displaystyle g(mn^{-1})^{-1} = g(m^{-1}n) = g(m^{-1})g(n)=g(m)^{-1}f(n)\)
\(\displaystyle f(mn^{-1}) = f(m)f(n)^{-1} = g(m)f(n)^{-1}\)

Use these two and come to a conclusion.

5) I'm not sure what a "factorial domain" is. Is it the same as the Unique Factorization Domain, in which the fundamental theorem of arithmetic applies?

If so, write each element as a product of prime elements (grouped together by powers from least to greatest) from R and compare their powers.