# Finite Field Tools

## How to generate a finite field in Sage

To generate a field of size \$p^n\$, for \$p\$ a prime, input \$p\$ and \$n\$ in the first two lines of this box. The field will be called “R”.
The result will give you an element \$a\$ which is a multiplicative generator (the variable we write polynomials in, instead of \$x\$). It will tell you the irreducible polynomial in \$a\$ that was used to create the field.

## Do some arithmetic in your field

Wrapping polynomials in the variable \$a\$ with the field \$R\$ will let you do arithmetic.

## Create a finite field with a given irreducible polynomial

If you want to control the irreducible polynomial (instead of letting Sage do it), this box shows how.

## Addition Table for the finite field

The first two lines set up the prime and the polynomial.

## Multiplication Table for the finite field

The first two lines set up the prime and the polynomial.

## Create a polynomial ring over Z/pZ, and factor polynomials or check irreducibility

Other things you might want to do.