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Math 114 - Spring 2003 - Borcherds - Final

# Math 114 - Spring 2003 - Borcherds - Final - MON 17:36 FAX...

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Unformatted text preview: 09/01/2003 MON 17:36 FAX 6434330 MOFFITT LIBRARY .001 Math 114. Final 2003 May 23. R. smoky-295\$ Please make sure that your name is on everything you hand in. You are allowed calculators and 1 sheet of notes. All questions have about the same number of marks. 1. Find polynomials (1(33), 5(33) in QM] such that (9:4 + 3:)a.(:17) + (562 + 1)b(:c) = 1 2. Prove that there exist irreducible polynomials over Q of arbitrarily large degree. (One way to do this is to use Eisenstein’s criterion.) 3. If a: is a root of 2:3 — 33: + 1 show that 1/(1 —— or) is also a root. Use this to show that over any ﬁeld 35:3 — 333 + 1 is either irreducible or splits into the product of 3 linear factors. What is the Galois group of this polynomial over Q? 4. Find the Galois group of the splitting ﬁeld of. 334 — 2 over Q. What is its order? Is it abelian? Is it cyclic? 7 5. Give an example of a ﬁeld of characteristic p > 0 such that the Frobe— nius automorphism a: :—> mp is not onto. 6. Prove that the Galois group of GF(p”) : GF(p) is cyclic of order n, I and describe a generator of it. (GF(p”) is the ﬁnite ﬁeld of order 39”.) 7. Find the conjugacy classes of the dihedral group D10 of order 10. 8. What is the Galois group of 2:7 — 1 over Q? 9. Construct a ﬁeld with 16 elements. 10. Prove that the additive group of any ﬁnite ﬁeld of order p” (p prime) is a product of n cyclic groups of order p. ...
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