Tag Archives: sage

A little intro to the Jacobi symbol: Part 1

If $m$ and $n$ are relatively prime integers, the Jacobi symbol $(m/n)$ is defined as the sign of the permutation $x\mapsto mx$ on the set $\Z/n$. Let’s give a simple example: $(7/5)$. The permutation on $\{1,2,3,4\}$ is given by $(1 2 4 3) = (1 2)(2 4)(4 3)$ which has an odd number of transpositions. […]

Polynomial over finite field: permutation polynomial?

Let’s assume you have a polynomial over a finite field $\F_q$, defined in Sage. How can you tell whether it’s a permutation polynomial? That is, when is the corresponding function $\F_q\to\F_q$ bijective? This is how you might have a polynomial over $\F_q$ defined in Sage:

Here, the variable $x$ refers the element $x$ in […]

Working with group rings in Sage

Let $\Z[\Z/n]$ denote the integral group ring of the cyclic group $\Z/n$. How would you create $\Z[\Z/n]$ in Sage so that you could easily multiply elements? First, if you’ve already assigned a group to the variable ‘A’, then

will give you the corresponding group ring and store it in the variable ‘R’. The first […]

How to Add a Table of Contents to a Sage Worksheet

Sage is a neat bundle of mathematical software that can be used to do anything from finding class numbers of number fields (like I did to make this graph) to testing whether a finitely presented group is trivial (sometimes…). Typically one works with Sage in a “worksheet” in a web browser, so that it’s easy […]