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Unformatted text preview: d Z , we have h a,b i h d i . 3. I = { 2 r + sx  r,s Z [ x ] } . If f ( x ) I , f (0) = 2 r + s 0 = 2 r 0 mod 2. So, I { f ( x ) Z [ x ]  f (0) 0 mod 2 } . For the other direction, note that for f ( x ) to be in { f ( x ) Z [ x ]  f (0) 0 mod 2 } , implies that the constant term is even, so f ( x ) = 2 r + x h ( x ), where h ( x ) is any polynomial.Thus, we get { f ( x ) Z [ x ]  f (0) 0 mod 2 } I . 4. Since 2 I , the only way for I to be principal is if I = h 2 i (impossible since x / h 2 i ) or I = h 1 i (impossible since 1 / I ). 2...
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This note was uploaded on 09/05/2011 for the course MATH 3360 taught by Professor Billera during the Spring '08 term at Cornell University (Engineering School).
 Spring '08
 BILLERA
 Math, Algebra

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