Exploring Abstract Algebra with Mathematica

Al Hibbard, Central College
and
Ken Levasseur, UMass Lowell

This page illustrates the quadratic and cubic extensions of the rings Z2 and Z3.

  1. Select the ring you want to look at::

  2. Enter coefficients of the modulus polynomial. If you selected Z3 for your base ring, the polynomial should not be cubic.

    x3+ x2+ x+ = p(x)

Results

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Base Ring

R=

Modulus Polynomial

p(x)=

Quotient ring


Tables of operations

Created by webMathematica

As with all rings, the additive group is abelian. In this case, the additive group is isomorphic to the ring Rr+1, where r is the degree of p(x).

The quotient ring is a field if the modulus polynomial is irreducible. The presence of zerodivisors indicates that the modulus polynomial is not irreducible.

Created by webMathematica Created by webMathematica