I am reading Dummit and Foote on algebraic extensions. I am having some issues understanding Example 2 on page 526 - see attachment.

Example 2 on page 526 reads as follows:

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(2) Consider the field generated over by and .

Since is of degree 2 over the degree of the extension is at most 2 and is precisely 2 if and only if is irreducible over . ... ... etc etc

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My question is: whydoes it follow that the degree of the extension is at most 2 and is precisely 2 if and only if is irreducible over ?exactly

Although I may be being pedantic I also have a concern about why exactly is of degree 2 over . I know it is intuitively the case or it seems the case that the minimal polynomial is in this case but how do we demonstrate this for sure - or is it obvious? (I may be overthinking this??)

Can someone help with the above issues/problems?

Peter