Unformatted text preview: V be the set of all polynomials over Q of degree 3, together with the zero polynomial. Is V a vector space? Why or why not? 6. Recall that a ring homomorphism is one that preserves the operations of a ring, and that its kernel is an ideal. Deﬁne a vector space analogue of a ring homomorphism, and show that its kernel is a subspace. 7. Show that Q ( i, √ 2) = Q ( i + √ 2 , 4√ 32). 8. Is the extension Q ( π + 1) of Q algebraic or transcendental? Is the extension Q ( π ) algebraic or transcendental over the ﬁeld Q ( π 3 )? (Note that Q ( π 3 ) is indeed a subﬁeld of Q ( π ).) 1...
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 Fall '09
 Algebra, Vector Space, Prof. Jones

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