# Hmwk4 - From Section 3 of the lecture notes 1(15 pts Find the splitting ﬁelds over Z 2 for the following polynomials(a x 2 x 1(b x 2 1(c x 3 x

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Math 523 Homework 4 Due 7/21/2008 Total: 40 points From Section 2 of the lecture notes: In these exercises, let F be a ﬁeld, let p ( x ) be a nonconstant polynomial in F [ x ], and let R = F [ x ] / h p ( x ) i . 1. (5 pts) If f ( x ) is any polynomial, show that h [ f ( x )] i = h [ d ( x )] i in R , where d ( x ) = gcd( f ( x ) , p ( x )). 2. (5 pts) Suppose that p ( x ) = g ( x ) h ( x ), where gcd( g ( x ) , h ( x )) = 1. Show that in R we have h [ g ( x )] i = { [ f ( x )] R | f ( x ) h ( x ) 0 (mod p ( x )) } . 4. (5 pts) In Z 2 [ x ] / ± x 15 - 1 ² , ﬁnd the idempotent generator for the ideal ± x 4 + x + 1 ² . Be sure to show that your answer is in fact idempotent.
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Unformatted text preview: From Section 3 of the lecture notes: 1. (15 pts) Find the splitting ﬁelds over Z 2 for the following polynomials: (a) x 2 + x + 1 (b) x 2 + 1 (c) x 3 + x + 1 (d) x 3 + x 2 + 1 2. (5 pts) Find the splitting ﬁeld for x p-x over Z p . 3. (5 pts) Show that if F is an extension ﬁeld of K of degree 2, then F is the splitting ﬁeld over K for some polynomial....
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## This note was uploaded on 06/13/2011 for the course CRYPTO 101 taught by Professor Na during the Spring '11 term at Harding.

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