# The Existence of Multiplicative Inverses

by Ricardo Alfaro (University of Michigan - Flint) and Steven Althoen (University of Michigan - Flint)

College Mathematics Journal
May, 2006

Subject classification(s): Algebra and Number Theory | Linear Algebra
Applicable Course(s): 4.2 Mod Algebra I & II

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.

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