Math787_HW5 - p = { f ( x ) A [ x ] | f (0) m } . Show that...

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Math 787: Preparation for Algebra Qualifying Exam, Summer 2008. Homework V: Field and Ring Theory 1. Let F be a field with n elements and let f ( x ) , g ( x ) ±F [ x ] so that f ( x ) 6 = g ( x ) and f ( a ) = g ( a ) for all a±F . Show that deg ( f ( x ) - g ( x ) n . 2. Let m be a maximal ideal of a commutative ring A with unit. Let
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Unformatted text preview: p = { f ( x ) A [ x ] | f (0) m } . Show that p is a maximal ideal of A [ x ]. 3. Show that x 4 + 1 is reducible in Z /p [ x ] for all primes p . 4. Find deg F ( 2 + 3) when F = Q and also when F = Q ( 6). 1...
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