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Algebra Jan 2007

# Algebra Jan 2007 - where the p i are distinct primes Show...

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Algebra Qualifying Exam January 2007 1. Let G be a group of order 21. Show that there is an element x G of order 7 such that ( x ) G . 7 pts. 2. Let G be a finite nilpotent group. Let M be a maximal proper subgroup of G . Show that the index of M in G is prime. 7 pts. 3. (a) Define the cyclotomic polynomial φ p ( x ) Z [ x ] where p is a prime number and prove that it is irreducible over Q . (b) Is x 5 + x 4 + x 3 + x 2 + x + 1 irreducible over Q ? 10 pts. 4. (a) Let F E be a field extension and α E an algebraic element over F . Define irr ( α, F ) (in any of the various equivalent ways). (b) Let α = radicalbig 1 3 C . Find irr ( α, Q ). 10 pts. 5. Let F be an arbitrary field. (a) State and prove an equivalent condition for a polynomial f ( x ) F [ x ] to be such that f ( x ) = 0. (b) Let p ( x ) F [ x ] be an irreducible polynomial. Prove that p ( x ) has multiple roots p ( x ) = 0 . (c) Can you give an example of an irreducible polynomial p ( x ) Q [ x ] with multiple roots? 14 pts. 6. Let G be a finite group. For each n Z + , let T n ( G ) := { x G : x n = 1 } . Suppose that for every divisor n of | G | we have that | T n ( G ) | ≤ n . (a) Let | G | = p a 1 1 · · · p a r r where the
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Unformatted text preview: where the p i are distinct primes. Show that T p a i i ( G ) is the unique p i-Sylow of G . (b) Show that T p a i i ( G ) is cyclic. (c) Deduce that G is cyclic. (d) Let F be a Fnite Feld. Show that F \ { } is a cyclic group under multiplication. 20 pts. 7. Let R be a commutative ring with 0 n = 1. Let P be a prime ideal of R and S := R \ P . (a) Show that S is a submonoid of ( R, · ). (b) Consider the ring of fractions S − 1 R . Let I := { x s ∈ S − 1 R : x ∈ P, s ∈ S } . Show that I is the unique maximal ideal of S − 1 R . 12 pts. 8. (a) DeFne the notion of projective module. (b) Let d be a divisor of n . Describe a nontrivial Z n-module structure on Z d . Explain why it is well-deFned. (c) Suppose gcd( d, n/d ) = 1. Show that the Z n-module Z d is projective. (d) Show that the Z 4-module Z 2 is not projective. 20 pts. 1...
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