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)?

Solution PreviewSolution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden.

Abstract Algebra Problems
    $25.00 for this solution

    PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

    Find A Tutor

    View available Abstract Algebra Tutors

    Get College Homework Help.

    Are you sure you don't want to upload any files?

    Fast tutor response requires as much info as possible.

    Upload a file
    Continue without uploading

    We couldn't find that subject.
    Please select the best match from the list below.

    We'll send you an email right away. If it's not in your inbox, check your spam folder.

    • 1
    • 2
    • 3
    Live Chats