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 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

 

[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