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Unformatted text preview: Math 4108 Study Sheet for Midterm 1 February 23, 2010 1. Know the books definition of prime elements, irreducible elements of polynomial rings. Know how to show that if p is a prime element of a PID, and p | ab , then p | a or p | b (on the exam, I will tell you which version of the definition of prime element to use). Know how to compute gcds in Euclidean Domains, in particular in F [ x ], where F is a field. 2. Know the fact that fields have only trivial ideals. Know that if f is a non-trivial homomorphism out of a field, then f must be injective. Know the fact that if D is a domain, and I is a maximal ideals of D , then D/I is a field. Know the characteristic of a domain. Know the fact that every finite integral domain is a field. 3. Understand the proof that if D is a Euclidean Domain, then D is also a UFD; also, know the fact that this holds more generally for PID. In particular, know how to show F [ x ] is a ED, where F is a field; so, F [ x ] is a UFD. Know the obvious example showing the Z [ x ] is not a PID (The example is I is the set of all polynomials with an even...
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