Abstract Algebra: Theory and Applications · Problem SE-2

Polynomials — Thomas Judson

Mathematics — Abstract Algebra · ideals-sage

Problem

Define

R

to be the ring of integers,

\mathbb{Z}

, by executing R = ZZ or R = Integers(). A command like R.ideal(4) will create the principal ideal

\langle 4 \rangle

. The same command can accept more than one generator, so for example, R.ideal(3, 5) will create the ideal

\{a \cdot 3 + b \cdot 5 \mid a, b \in \mathbb{Z}\}

. Create several ideals of

\mathbb{Z}

with two generators and ask Sage to print each as you create it. Explain what you observe and then create code that will test your observation for thousands of different examples.

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