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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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- Fall '11