Adv Alegbra HW Solutions 111

Adv Alegbra HW Solutions 111 - p 1 ( x ), . . . , p m ( x )...

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111 Solution. Absent. (ii) Find two different ways of grouping the factors of 65 in part (i) as a product α α of two conjugate factors. Use these to get two different expressions for 65 as a sum of two squares in Z . Solution. Absent. 3.81 Prove that there are domains R containing a pair of elements having no gcd. Solution. Let k be a f eld, and let R be the subring of k [ x ] consisting of all polynomials having no linear term; that is, f ( x ) R if and only if f ( x ) = s 0 + s 2 x 2 + s 3 x 3 +··· . We claim that x 5 and x 6 have no gcd: their only monic divisors are 1 , x 2 , and x 3 , none of which is divisible in R by the other two. For example, x 2 is not a divisor of x 3 , for if x 3 = f ( x ) x 2 , then (in k [ x ] )wehavedeg ( f ) = 1. But there are no linear polynomials in R . 3.82 True or false with reasons. (i) Every element of Z [ x ] is a product of a constant in Z and monic irreducible polynomials in Z [ x ] . Solution. False. (ii) Every element of Z [ x ] is a product of a constant in Z and monic irreducible polynomials in Q [ x ] . Solution. True. (iii) If k is a f eld and f ( x ) k [ x ] can be written as ap 1 ( x ) ··· p m ( x ) and bq 1 ( x ) ··· q n ( x ) , where a , b are constants,
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Unformatted text preview: p 1 ( x ), . . . , p m ( x ) are monic irreducible polynomials, and q 1 ( x ), . . . , q n ( x ) are monic nonconstant polynomials, then q 1 ( x ), . . . , q n ( x ) are irreducible. Solution. False. (iv) If k is a f eld and f ( x ) k [ x ] can be written as ap 1 ( x ) p m ( x ) and bq 1 ( x ) q n ( x ) , where a , b are constants, p 1 ( x ), . . . , p m ( x ) are monic irreducible polynomials, and q 1 ( x ), . . . , q n ( x ) are monic nonconstant polynomials, then m n . Solution. True. (v) If k is a sub f eld of K and f ( x ) k [ x ] has the factorization f ( x ) = ap e 1 1 p e n n , where a is a constant and the p i ( x ) are monic irreducible in k [ x ] , then f ( x ) = ap e 1 1 p e n n is also the factorization of f ( x ) in K [ x ] as a product of a constant and monic irreducible polynomials. Solution. False....
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This note was uploaded on 12/21/2011 for the course MAS 4301 taught by Professor Keithcornell during the Fall '11 term at UNF.

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