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I am currently focused on Chapter 7: Field Extensions ... ...

I need help with an aspect of the proof of Theorem 7.1.10 ...

Theorem 7.1.10, and the start of its proof, reads as follows:

View attachment 6575

In the above text from Lovett we read the following ...

" ... ... Let \(\displaystyle p(x)\) be a polynomial of least degree such that \(\displaystyle p( \alpha ) = 0\) ... ... "

Then Lovett goes on to prove that \(\displaystyle p(x)\) is irreducible in \(\displaystyle F[x]\) ... ...

... BUT ... I am confused by this since it is my understanding that if \(\displaystyle p( \alpha ) = 0\) then \(\displaystyle p(x)\) has a linear factor \(\displaystyle x - \alpha\) in \(\displaystyle F[x]\) and so is not irreducible ... ... ?

Can someone please help clarify this issue ... ...

Peter