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practice_midterm_2_problems

practice_midterm_2_problems - V be the set of all...

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College of the Holy Cross, Fall 2009 Math 352, Practice Midterm 2 1 Prof. Jones 1. Find the splitting ﬁeld of f ( x ) = x 3 + x + 1 Z 2 [ x ] over Z 2 . 2. (a) The extension Q ( 6 5) is a vector space over Q . Find a basis for it. (b) Let α be a root of the polynomial f ( x ) = x 3 + x + 4 Z 5 [ x ]. Find a basis for the vector space Z 5 ( α ) over Z 5 . 3. Suppose that D is a Euclidean Domain (recall that this means D has a function called a measure that makes D have a division algorithm – see p. 329 for details). Let I be a non-zero ideal of D , and let α I be such that α has minimal measure among non-zero elements of D . Prove that I = h α i . 4. How many elements are in the ring Z [ 2] / h 2 i ? Use your answer to show that h 2 i is a maximal ideal in Z [ 2]. 5. Let
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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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