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e3sol - 1 Math 417 Exam III Solutions August 5 2009 1 Let k...

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Unformatted text preview: 1 Math 417 Exam III Solutions: August 5, 2009 1 . Let k be a field and let f ( x ) ∈ k [ x ] be a polynomial of degree n . Prove that f ( x ) has at most n roots in k . This is Theorem 3.50 in the book. 2 . Prove that F = { a + b √ 2 : a, b ∈ Q } is a field. It suffices to prove that F is a subring of R (for subrings are rings in their own right) and that every nonzero element has an inverse in F . Now F is a subring if 1 ∈ F and F is closed under subtraction and multipli- cation. First, 1 = 1+0 i ∈ F . Second, ( a + b √ 2)- ( c + d √ 2) = ( a- c )+( b- d ) √ 2, which is in F because a- c, b- d ∈ Q . Third, ( a + b √ 2)( c + d √ 2) = ( ac + 2 bd ) + ( ad + bc ) √ 2 ∈ F. We conclude that F is a commutative ring. Finally, assume that a + b √ 2 6 = 0. The equation ( a + b √ 2)( a- b √ 2) = a 2- 2 b 2 gives ( a + b √ 2)- 1 = a a 2- 2 b 2 + b a 2- 2 b 2 √ 2 ....
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e3sol - 1 Math 417 Exam III Solutions August 5 2009 1 Let k...

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