Fall 13 Math 410 Final

Math 410, Galois Theory, Fall 2013
Final Exam (due on Wednesday 18 December by 5pm)
• You should explain all your answers clearly and completely in order to get
full credit.
• You may not collaborate on or discuss the problems with anyone except me.
• You may refer to the textbook, your notes from class, your corrected problem sets, but
no other books, notes or outside sources (including the internet).
• In answering the questions, you may use (without proof) any result stated in class or
in the main body of the textbook sections that we have covered, or any result from a
homework exercise, unless otherwise instructed. (You may not use results of exercises
in the textbook that were not assigned for homework, or results from sections of the
textbook that we did not cover.) You should make it clear what results you are referring
to (by name, number, or by restating the result).
• If you are unsure what you are allowed to use, what level of detail is required, or
whether you should explain something, please ask me.
Questions
1. Prove that every degree two field extension is normal.
2. Find, with justification, all the roots of the minimal polynomial of
√
4
3+
√
6
3 over Q.
3. Describe all the intermediate fields of the cyclotomic extension Q ⊆ Q(ζ9 ). You should
give (with proof) an explicit primitive element for each intermediate field as an extension of Q.
4. Let n be a positive integer. Show that C(tn ) ⊆ C(t) is a Galois extension and calculate
its Galois group.
5. Let f ∈ Q[x] be an irreducible polynomial of degree 5 with Galois group isomorphic
to C5 . Prove that all the roots of f in C are real.
6. Show that the Galois group of x8 − 2 ∈ Q[x] has 16 elements including an element
of order 8. (Challenge, not for credit: is this Galois group isomorphic to the dihedral
group D16 ?)
7. Let p be a prime. Is the extension Fp ⊆ Fpp solvable?
8. Let p be a prime and f ∈ Fp [x] an irreducible polynomial of degree d.
(a) Prove that f is irreducible in Fpn [x] if and only if gcd(d, n) = 1.
(b) If gcd(d, n) = k, prove that f factors in Fpn [x] into k irreducible factors, each of
degree kd .

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