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I am reading Abstract Algebra: Structures and Applications" by Stephen Lovett ...

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:

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

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

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

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

Peter

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:

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

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

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

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

Peter