# The Existence of Multiplicative Inverses

Using only basic ideas from linear algebra and number theory, the authors show that if $c$ is square-free, the ring $Q [\sqrt[n]{c}]$ is a field. An arbitrary nonzero element of the ring is associated with a system of equations, and divisibility arguments are used to show that a matrix of coefficients from the system must have a nonzero determinant, eventually leading to the result that the original element of the ring has an inverse.

Old Node ID:
1352
Author(s):
Ricardo Alfaro (University of Michigan - Flint) and Steven Althoen (University of Michigan - Flint)
Publication Date:
Monday, December 11, 2006
Original Publication Source:
College Mathematics Journal
Original Publication Date:
May, 2006
Subject(s):
Algebra and Number Theory
Linear Algebra
Topic(s):
Linear Algebra
Determinants
Matrix Algebra
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File Content:
Rating Count:
80.00
Rating Sum:
218.00
Rating Average:
2.73
Author (old format):
Ricardo Alfaro and Steven Althoen
Applicable Course(s):
4.2 Mod Algebra I & II
Modify Date:
Monday, December 11, 2006