Field Extensions - Lovett, Theorem 7.1.10 .... ....

[Math Amateur]

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:
View attachment 6575In 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

 

[Euge]

If $\alpha\notin F$, $x - \alpha$ is not a polynomial in $F[x]$, so it cannot be a linear factor of $p(x)$ in $F[x]$. The condition $[F(\alpha):F] > 1$ holds if and only if $\alpha \in F$.

 

[Math Amateur]

If $\alpha\notin F$, $x - \alpha$ is not a polynomial in $F[x]$, so it cannot be a linear factor of $p(x)$ in $F[x]$. The condition $[F(\alpha):F] > 1$ holds if and only if $\alpha \in F$.

Thanks for the help, Euge

Peter