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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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Publish Page: 
Furnished by JSTOR: 
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
Average: 2.8 (80 votes)

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