Math787_HW4 - 3 Show that f x = x 2 1 ± Z 3 x is...

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Math 787: Preparation for Algebra Qualifying Exam, Summer 2008. Homework IV: Ring Theory 1. Show that Z [ i ] is a Euclidean domain with ν ( x ) = a 2 + b 2 if x = a + bi . Show that 3 ± Z [ 5 i ] is irreducible, but not prime. 2. Find a prime ideal of Z 16 [ x ] that is not a maximal ideal.
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Unformatted text preview: 3. Show that f ( x ) = x 2 + 1 ± Z 3 [ x ] is irreducible, but x 4 + 1 is reducible. 4. Let U be the Abelian group of units of a finite field. Show that U is cyclic. 1...
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