Unformatted text preview: A is a normal lower triangular matrix over the complex numbers, the A is a diagonal matrix. 6. Let R = Z [ X ] and let I = (2 ,X ). Prove that I is not a free Rmodule but it is torsion free. 7. Let F be a ﬁnite ﬁeld. Prove that the product of the non zero elements of F is1. 8. Let ξ = p 2 + √ 2. Find the minimal polynomial of ξ over Q and show that ξ = p 2√ 2 is another root of this minimal polynomial. Show that the degree of Q ( ξ ) over Q is 4. Prove that sending ξ to ξ = p 2√ 2 is an automorphism of Q ( ξ ) over Q . Describe the Galois group of Q ( ξ ) over Q . 1...
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This note was uploaded on 06/19/2011 for the course MATH 699 taught by Professor Na during the Spring '11 term at University of North Carolina School of the Arts.
 Spring '11
 NA
 Algebra

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