ALGEBRA PRELIMINARY EXAM JANUARY 1995

(1) Let G be the dihedral group generated by elements a and b with relations

a^{6} = 1, b^{2} = 1, ba = a^{5}b.

(i) Find the order of G by showing that each element of G can be written uniquely in the form a^{i}b^{j}.

(ii) Prove that the subgroup generated by a is normal. (iii) Find the center of G. (iv) Find the centralizer of b in G.

(v) Find the number of conjugates of b in G.

(2) Prove that there is no simple group of order 80.

(3) Let R = F [x, y], the polynomial ring in two variables over the field F. Let

I be the ideal in R generated by the two elements x and y.

(i) Prove that any two elements of the R-module I are linearly dependent over R.

(ii) Prove that I is not a free R-module.

(4) Let J be the ideal (3, x^{3} − x + 1) in ℤ[x]. Prove that J is a prime ideal. Is J a maximal ideal?

Typeset by AMS-TEX

(5) Give an example of an integral domain that is not a unique factorization domain.

(6) Let p be a prime and let 𝔽_{p} denote the field with p elements. Let f(x) = x^{p^{n}} − x in 𝔽_{p}[x] for an integer n ≥ 1. Let K denote the splitting field of f(x) over 𝔽_{p}.

(i) Prove that K has p^{n} elements.

(ii) Prove that there is an element α in K such that K = 𝔽_{p}(α).

(iii) Prove that there is an irreducible polynomial of degree n in 𝔽_{p}[x].

(7) Let K/F be a finite Galois extension, and let L be an intermediate field, F ⊆ L ⊆ K. Let G be the group of automorphisms of K that fix the elements of F and let H be the group of automorphisms of K that fix the elements of L. Prove that if L/F is a Galois extension, then H is a normal subgroup of G.

(8) Let R be a ring with identity and let M be a left R-module. Prove that if f is a surjective R-homomorphism of M onto R, then M ≅ Ker(f) ⨁ R.