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I am reading John Fraleigh's book, A First Course in Abstract Algebra.
I am at present reading Section 22: Rings of Polynomials.
I need some help with an aspect of Fraleigh's discussion of "solving a polynomial equation" or "finding a zero of a polynomial" ...
The relevant text in Fraleigh is as follows:View attachment 4560
In the above text, we read the following:
" ... ... In terms of this definition, we can rephrase the classical problem of finding all real numbers r such that \(\displaystyle r^2 + r - 6 = 0\) by letting \(\displaystyle F = \mathbb{Q}\) and \(\displaystyle E = \mathbb{R}\) and finding all \(\displaystyle \alpha \in \mathbb{R}\) such that
\(\displaystyle \phi_\alpha ( x^2 + x - 6 ) = 0
\)
that is finding all zeros of \(\displaystyle x^2 + x - 6\) in \(\displaystyle \mathbb{R}\) ... ... "My question is as follows:
What is the relevance of the field \(\displaystyle F\)? It appears that if we made \(\displaystyle F = \mathbb{R}\) we would have achieved the same result ... .. indeed (if we regard a field as a subfield of itself) we could have taken \(\displaystyle F = E\) and achieved the same result ...
Can someone please explain the relevance of the subfield \(\displaystyle F\)? ... ... I am sure that I am missing something ...
Peter
*** NOTE ***I do understand that changing \(\displaystyle F\) changes the nature/type of the polynomials that can be input to the homomorphism \(\displaystyle \phi_\alpha\) since the co-efficients of the polynomial come from \(\displaystyle F\) ... but still do not really see the point or relevance of the subfield \(\displaystyle F\) ...
I am at present reading Section 22: Rings of Polynomials.
I need some help with an aspect of Fraleigh's discussion of "solving a polynomial equation" or "finding a zero of a polynomial" ...
The relevant text in Fraleigh is as follows:View attachment 4560
In the above text, we read the following:
" ... ... In terms of this definition, we can rephrase the classical problem of finding all real numbers r such that \(\displaystyle r^2 + r - 6 = 0\) by letting \(\displaystyle F = \mathbb{Q}\) and \(\displaystyle E = \mathbb{R}\) and finding all \(\displaystyle \alpha \in \mathbb{R}\) such that
\(\displaystyle \phi_\alpha ( x^2 + x - 6 ) = 0
\)
that is finding all zeros of \(\displaystyle x^2 + x - 6\) in \(\displaystyle \mathbb{R}\) ... ... "My question is as follows:
What is the relevance of the field \(\displaystyle F\)? It appears that if we made \(\displaystyle F = \mathbb{R}\) we would have achieved the same result ... .. indeed (if we regard a field as a subfield of itself) we could have taken \(\displaystyle F = E\) and achieved the same result ...
Can someone please explain the relevance of the subfield \(\displaystyle F\)? ... ... I am sure that I am missing something ...
Peter
*** NOTE ***I do understand that changing \(\displaystyle F\) changes the nature/type of the polynomials that can be input to the homomorphism \(\displaystyle \phi_\alpha\) since the co-efficients of the polynomial come from \(\displaystyle F\) ... but still do not really see the point or relevance of the subfield \(\displaystyle F\) ...
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