MATH 421 Midterm: MATH 421 NIU Test2sample
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Exam ii: beachy, 4/16/2004, (20 points) complete 4 of the following 5 de nitions, a commutative ring r with identity is called an integral domain if, let r be a commutative ring. Then: (eisenstein"s irreducibility criterion) let f (x) = anxn + . + a0 be a polynomial with integer coe cients. Then f (x) is irreducible over the eld of rational numbers if: let i be a proper ideal of the commutative ring r with identity 1. Then r/i is an integral domain if and only if: let i be a proper ideal of the commutative ring r with identity 1. Then r/i is a eld if and only if: let d be an integral domain. Then in the factor ring f [x]/ hp(x)i, every nonzero element is invertible: (15 points) let r be a commutative ring with identity 1. Let i and j be ideals of r, with i j r. de ne.
