Quadratic Polynomials and Irreducibles and Primes

[Math Amateur]

I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ...

I need some help with the proof of Theorem 1.2.2 ...

Theorem 1.2.2 reads as follows:
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In the above text from Alaca and Williams, we read the following:

"... ... Then the roots of \(\displaystyle f(X)\) in \(\displaystyle F\) are \(\displaystyle -ds/p\) and \(\displaystyle -d^{-1} t \). But \(\displaystyle d^{-1} t \in D\) while neither \(\displaystyle a/p\) nor \(\displaystyle b/p\) is in \(\displaystyle D\). Thus no such factorization exists. ... I am unsure of how this argument leads top the conclusion that \(\displaystyle f(X)\) does not factor into linear factors in \(\displaystyle D[X]\) ... in other words how does the argument that "" ... \(\displaystyle d^{-1} t \in D\) while neither \(\displaystyle a/p\) nor \(\displaystyle b/p\) is in \(\displaystyle D\) ... "lead to the conclusion that no such factorization exists. ...

Indeed ... in particular ... how does the statement "neither \(\displaystyle a/p\) nor \(\displaystyle b/p\) is in \(\displaystyle D\)" have meaning in the assumed factorization \(\displaystyle f(X) = (cX + s) ( dX + t )\) ... ... ? ... What is the exact point being made about the assumed factorization ... ?I am also a little unsure of what is going on when Alaca and Williams change or swap between \(\displaystyle D[X]\) and \(\displaystyle F[x]\) ...Can someone help with an explanation ...

Help will be appreciated ...

Peter

 

[Euge]

Hi Peter,

The contradiction here is that $-d^{-1}t$ is in $D$ and also not in $D$. As $d, t\in D$, $-d^{-1}t\in D$. On the other hand, $-d^{-1}t$ is one of the roots of $f$ in $F[X]$, namely $a/p$ or $b/p$; since neither $a/p$ nor $b/p$ is in $D$, $-d^{-1}t$ is not in $D$.

 

[Math Amateur]

Hi Peter,

The contradiction here is that $-d^{-1}t$ is in $D$ and also not in $D$. As $d, t\in D$, $-d^{-1}t\in D$. On the other hand, $-d^{-1}t$ is one of the roots of $f$ in $F[X]$, namely $a/p$ or $b/p$; since neither $a/p$ nor $b/p$ is in $D$, $-d^{-1}t$ is not in $D$.

Thanks Euge ... that is clear now ...

Appreciate your help ...

Peter