# Abstract Algebra : Splitting Fields

## Example Questions

### Example Question #1 : Fields

What definition does the following correlate to?

If  is a prime, then the following polynomial is irreducible over the field of rational numbers.

Possible Answers:

Principal Ideal Domain

Primitive Field Theorem

Gauss's Lemma

Eisenstein's Irreducibility Criterion

Ideals Theorem

Correct answer:

Eisenstein's Irreducibility Criterion

Explanation:

The Eisenstein's Irreducibility Criterion is the theorem for which the given statement is a corollary to.

The Eisenstein's Irreducibility Criterion is as follows.

is a polynomial with coefficients that are integers. If there is a prime number  that satisfy the following,

Then over the field of rational numbers  is said to be irreducible.