simple field extension

Definition

A field extension is called simple if where is the smallest subfield of which contains and .

Evaluation map and degree

Let be a simple extension. Given the evaluation homomorphism

1. is injective if and only if the degree , in which case

where denotes the field of fractions of .

2. is non-injective if and only if , in which case there exists a unique monic irreducible polynomial of degree , (call it ) such that

moreover, is the minimal polynomial (polynomial of least degree which satisfies ).

Proof

Part 1

Say Since is a field, the only non-unit is , so thus every unit in is mapped to a unit in under , therefore the universal property of fraction fields applies, and we have the following diagram

From this diagram, we can see that is injective and both and (which contains all generators of ) are in the image of . Therefore, is an isomorphism of fields, which is .

This gives

On the right hand side, is all rational functions of which is a subspace and since is a basis. Therefore, .

Part 2

Assume . By the first isomorphism theorem,

We know that is a field, so it is also an integral domain, therefore is also an integral domain. This implies is a prime ideal.

is a PID, which means that for some monic polynomial (we can assume is monic because is a field) which vanishes on . Since is prime, then it is also irreducible.

This gives the isomorphism

We can follow the implications

is then a subfield of that contains and so . Note that when the degree of is since it has a basis of .

Lastly, we know that is minimal since by definition so and since this ideal is principal so by the division algorithm there is none of smaller degree.

Importance of theorem

Simple extensions are (in some way) concrete. They are either quotients of polynomial rings (by some specific polynomial) or they are fraction field which are rational functions. Both of these constructions are more understandable than general fields.