Fields

We will now be exploring Fields in SageMath. Recall that there is one field of order $p^k$ where $p$ is a prime, that the Galois field of order $p^k$. In SageMath this is defined by writing $GF(p^k)$. Recall that we can also come up with these fields taking a quotient with a maximal ideal of a ring.

There are a few other functions that we will be using to study Fields, namely

• .modulus() - returns the modulus of the field.
• .multiplicative_order() - returns the multiplicative order of the field.
• factor() - factors an element of the field, given it is not equivalent to zero.

Galois Groups

Studying Galois groups in SageMath will require a few new functions and classes that we have not yet encountered, namely:

• QuadraticField($a$) - forms a quadratic extension of $\mathbb{Q}$ by adjoining $\sqrt{a}$ to the rational numbers.
• .galois_group() - returns the Galois group of a field.
• NumberField($p(x)$) - forms a field extension over $\mathbb{Q}$ by adjoining the roots of $p$ to $\mathbb{Q}$.

We go over an example in the next cell.

Note the difference in syntax between these two different methods, when using the NumberField() constructor, we must specify the generator, or SageMath returns an error.