I am reading "Introductory Algebraic Number Theory"by Saban Alaca and Kenneth S. Williams ... and am currently focused on Chapter 1: Integral Domains ...(adsbygoogle = window.adsbygoogle || []).push({});

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

Theorem 1.2.2 reads as follows:

In the above text from Alaca and Williams, we read the following:

"... ... Then the roots of ##f(X)## in ##F## are ##-ds/p## and ##-d^{-1} t ##. But ##d^{-1} t \in D## while neither ##a/p## nor ##b/p## is in ##D##. Thus no such factorization exists. ...

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

Indeed ... in particular ... how does the statement "neither ##a/p## nor ## b/p## is in ##D##" have meaning in the assumed factorization ##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 ##D[X]## and ##F[X]## ...

Can someone help with an explanation ...

Help will be appreciated ...

Peter

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# I Quadratic Polynomials and Irreducibles and Primes ...

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