# Abstract Algebra Problems

## Question

1. Show that any homomorphism ψ of one field onto another is an isomorphism (Hint: It suffices to show that ψ cannot send a nonzero element to 0).

2. Show that six combinations of values of σ(3√2) and σ(ζ3) are possible for an isomorphism of Q(3√2, ζ3) onto itself, and that all six actually occur for composites of the isomorphisms.

3. Under what conditions is Cₘ × Cₙ cyclic?

4. If G is any finite group and g ∈ G, show that the powers 1, g, g², … form a subgroup of G. (The size of this subgroup, which is the least n such that gⁿ = 1, is called the order of g).

5. how that A4 has no subgroup with six elements (which shows, incidentally, that the converse of Language’s theorem is false)

6. What relationships between automorphism groups are revealed by restricting the automorphisms of Q(√2, √3) to Q(√2)?

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