I am reading Nicholson: Introduction to Abstract Algebra, Section 6.2 - Algebraic Extensions.

Example 15 on page 282 (see attachment) reads as follows:

---------------------------------------------------------------------------------------------------------------------------------

Example 15.

Let .

Find , exhibit a -basis of E, and show that . Then find the minimum polynomial of over .

-----------------------------------------------------------------------------------------------------------------------------------

In the solution we read:

We write for convenience so that ... ... etcSolution:

... ... ... We claim that is the minimal polynomial of over L. Because and are the only roots of in , we merely need to show that . ... ... etc

My problem is the following:

How does showing imply that is the minimal polynomial of over L?

Can someone please help with this issue?

Peter