Advanced Algebra I
Homework 6 due on Nov. 10, 2006
(1) * Complete the uncompleted proof in the lecture.
(2) Construct a field F of 9 elements. Then F∗ is a cyclic group of 8 elements. Find a generator of F∗.
* How about fields of 3n elements?
(3) Let F = Q(√
3, i, ω), where ω = −1+2√3i. Find [F : Q] and a basis of F over Q.
(4) Let F = Q(√3
2, ω). Find [F : Q] and a basis of F over Q.
Moreover, find an element u such that F = Q(u).
(5) Verify Proposition 3.27.
(6) In the field K(x) we consider u = x4+xx+12+1 What is [K(x) : K(u)]? In general, if u = f (x)g(x), then what is [K(x) : K(u)]?
(7) Let Φp(x) := xx−1p−1 = xp−1+ ... + 1 ∈ Q[x]. Show that Φp(x) is irreducible.
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