Adv Alegbra HW Solutions 118

Adv Alegbra HW Solutions 118 - f eld k and if the quo-tient...

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118 3.92 True or false with reasons. (i) If I is a proper ideal in a commutative ring R and π : R R / I is the natural map, then ker π = I . Solution. True. (ii) If I is a proper ideal in a commutative ring R and π : R R / I is the natural map, then π is surjective. Solution. True. (iii) If f : R S is a homomorphism of commutative rings, then S has a subring isomorphic to R /( ker f ) . Solution. True. (iv) If I is a proper ideal in a commutative ring R , then R has a subring isomorphic to R / I . Solution. False. (v) If p is a prime number, then every f eld of characteristic p is f nite. Solution. False. (vi) Every f eld of characteristic 0 is in f nite. Solution. True. (vii) If f ( x ) is an irreducible polynomial over a f eld k , then k [ x ] /( f ( x )) is a f eld. Solution. True. (viii) If f ( x ) is a nonconstant polynomial over a
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Unformatted text preview: f eld k and if the quo-tient ring k [ x ] /( f ( x )) is a f eld, then f ( x ) is irreducible. Solution. True. (ix) If f ( x ) is an irreducible polynomial over a f eld k , then every ele-ment z k [ x ] /( f ( x )) is a root of f ( x ) . Solution. False. (x) If k K are f elds and z K is a root of some nonzero polyno-mial p ( x ) k [ x ] , then p ( x ) is irreducible in k [ x ] . Solution. False. (xi) There is a f eld containing C ( x ) and x + i . Solution. True. (xii) For every positive integer n , there exists a f eld with exactly 11 n elements. Solution. True....
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