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.