Finite Fields

## Description

Sage knows about finite fields. A finite field must have order $p^n$, where $p$ is a prime. We can denote a finite field of order $p^n$ by $\text{GF}(p^n)$. For example, we can construct the polynomial $2y^5 + 2y^4 + 4y^3 + 2y^2 + 3y + 1$ over a finite field of order $5$ and then check to see if this polynomial is irreducible.

## Sage Cell

#### Code

F = FiniteField(5)
S.<y> = F[]
p = 2*y^5 + 2*y^4 + 4*y^3 + 2*y^2 + 3*y + 1
p.is_irreducible()


## Options

#### Option

To construct a finite field of order $p^n$, where $n \geq 2$, we must specify a generator for the field.

#### Code

F.<a> = FiniteField(5^2)
F.list()


#### Option

The polynomial $2y^5 + 2y^4 + 4y^3 + 2y^2 + 3y + 1$ does factor over the field $\text{GF}(5^{15})$, a field of 30,517,578,125 elements.

#### Code

F.<a> = FiniteField(5^15)
S.<y> = F[]
p = 2*y^5 + 2*y^4 + 4*y^3 + 2*y^2 + 3*y + 1
p.factor()


## Tags

Primary Tags: abstract algebra

Secondary Tags: fields, finite fields, polynomials

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