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

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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:

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 ... ... ?

Peter

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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 $\alpha \in F$.

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MHB
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