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Unformatted text preview: Homework 10 Solutions 1. BWOC, suppose the highest degree irreducible polynomial in F [ x ] is n . Since F is finite, there are only a finite number of irreducible polynomials of each degree, and we dont have irreducible polynomials above degree n , thus our assumption implies that there are only a finite number of irreducible polynomials in F [ x ]. Enumerate these as p 1 ( x ) , . . . p m ( x ) (and m 2 clearly since x and x + 1 are irreducible in F [ x ]). Consider the polynomial f ( x ) = p 1 ( x ) p 2 ( x ) p m ( x ) + 1. deg ( f ( x )) > n so f ( x ) is reducible. Since p 1 , . . . p m are the only irreducible polynomials, one of these must divide f ( x ). p i | f ( x ) and p i ( x ) | p 1 ( x ) p 2 ( x ) p m ( x ) implies p i ( x ) | 1, a contradiction. Thus there must be irreducible polynomials of arbitrarily high degree. 2. (a) 5 x + 1 = (2 x + 1)(3 x + 1) (b) 5 x = (4 x + 3)(3 x + 2) (c) 2 x + 2 = (2 x + 2)(3 x + 1) (d) 2 x + 2 = (2 x + 2)(3 x 2 + 1) 3. (a) x 5 + 9 x 4 + 12 x 2 + 6 - Eisensteins Criterion with...
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