oct28 - (d) The binomial theorem shows that this polynomial...

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Math 5125 Friday, October 28 October 28, Ungraded Homework Exercise 9.4.2 on page 311 Prove that the following polynomials are irreducible in Z [ x ] : (a) x 4 - 4 x 3 + 6 (b) x 6 + 30 x 5 - 15 x 3 + 6 x - 120 (c) x 4 + 4 x 3 + 6 x 2 + 2 x + 1 (d) (( x + 2 ) p - 2 p ) / x , where p is an odd prime. (a) Apply Eisenstein for the prime 2. (b) Apply Eisenstein for the prime 3. (c) Define an automorphism of Z [ x ] by n 7→ n for n Z and x 7→ x - 1. An automorphism preserves irreducibility, so it will be sufficient to show that ( x - 1 ) 4 + 4 ( x - 1 ) 3 + 6 ( x - 1 ) 2 + 2 ( x - 1 )+ 1 is irreducible. This simplifies to x 4 - 2 x + 2. Now apply Eisenstein for the prime 2.
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Unformatted text preview: (d) The binomial theorem shows that this polynomial is of the form x p-1 + 2 px p-2 + + 2 p-1 p . The coefcients of x i for 0 i p-2 are all divisible by p ; it is crucial that p is prime here. Also p 2 does not divide 2 p-1 p (here it is crucial that p 6 = 2). Now apply Eisenstein for the prime p (of course, it is crucial that p is prime here)....
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