Pries: M467 - Abstract Algebra I, Spring 2011
Homework 1: Field extensions.
Be ready to present two problems by 2/16 and turn in seven problems by 2/18.
Read: Judson 19.1, Ash 3.1, Wilkons 3.1-3.2, etc.
Problems:
1. If n = wm, divide xn − 1 by xm − 1 and find the quotient and remainder.
2. Find the group of automorphisms Aut(Q) of Q.
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3. Show that 2 is irrational using this outline for a proof by contradiction:
Assume that there exist integers a, b with b 6= 0 so that 2 = (a/b)2 .
(i) Explain why we can suppose a and b are positive and gcd(a, b) = 1.
(ii) Using the equation a2 = 2b2 , explain why a must be even (and thus b odd).
(iii) Show that 4 divides a2 but not 2b2 , giving a contradiction.
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4. Suppose d and e are two integers. Under what conditions is Q( d) ' Q( e)?
5. Let r be a fixed positive real number. Describe the set of all z ∈ Q[i] with N (z) = r.
6. Let f (x) = x4 + 1.
(i) Factor f (x) over C.
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(ii) Find three quadratic fields K = Q( d) so that f (x) factors over K and write
down the factorizations.
(iii) Show that f (x) is irreducible over Q.
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7. Let α = 1 + 13. Find mα,Q(√13) (x). Find mα,Q (x).
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8. Let L = Q( 11, 2). Let α = 2 + 11. Prove that L = Q(α). Some steps are:
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(i) Show 11 6∈ Q( 2) and [L : Q( 2)] = 2.
(ii) Find [L : Q].
(iii) Find the minimal polynomial mα,Q (x).
(iv) Prove that L = Q(α).
9. Let z = ζ7 = e2πi/7 . Recall that z is a root of f (x) = x6 + x5 + x4 + x3 + x2 + x + 1.
Let α = z + z 2 + z 4 and let α0 = z 3 + z 5 + z 6 .
(i) Find t, n ∈ Q so that α + α0 = t and αα0 = n.
(ii) Find a polynomial g(x) ∈ Q[x] with roots α and α0 .
(iii) Solve for α and α0 using the quadratic formula.
10. Let M, K, L be fields with M ⊂ K ⊂ L. If the degree [L : M ] is prime, prove that
M = K or M = L.
11. Let L = Q(α) where the minimal polynomial mα,Q has degree 5. Prove that the
minimal polynomial of α2 over Q also has degree 5.