**Let be a field and be an indeterminate over . Let .Prove that is a field if and only of is irreducible.
**

Proof:

We know that for any ring , is a field if and only if is a maximal ideal of . To use it, we show that is maximal if and only if is irreducible.

Let be maximal. We show is irreducible. If possible, let where the degree of and is at least one, for some . Here, and so the ideals and are proper and also contains , which is a contradiction since is maximal. So the polynomial is irreducible.

Let be irreducible. We show is maximal. If possible, let for an ideal of . Since is a field so is a PID, so every ideal of is principal. Then for a polynomial is principal. Then for a polynomial , .

Since , so for there is some s.t. . Since is irreducible, so either or is of degree zero. If then is a unit in . Here, i.e. contains a unit, so . If then , for , constant. Then i.e. i.e. . Hence, . Which suggests that is maximal.

Now, let is a field, so the ideal is maximal. We have that if is maximal then is irreducible.

Let is irreducible then is maximal so is a field.