MAT 150C Study Guide - Midterm Guide: Galois Extension, Commutative Ring
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Bring the following integer matrix a to its canonical form d1 d2 d3 by multiplying elements of gl3(z) from the left, and elements of gl4(z) from the right, where the entries satisfy 0 < d1, and d1|d2|d3. Identify the structure of the z-module v presented by a : z4 z3. This problem concerns basic de nitions. (1) [3 points] let r be a commutative ring with 1. Give the de nition of a nitely generated r-module v . (2) [3 points] let f be a eld and f (x) f [x] a polynomial. State the de nition of a (3) [4 points] let k be a nite extension eld of another eld f . State the de nition splitting eld of f (x) over f . that k is a galois extension of f . Let a and b be two positive integers that are relatively prime. Prove that as a z-module, z/(a) z/(b) and z/(ab) are isomorphic:
