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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 $$r^2 + r - 6 = 0$$ by letting $$F = \mathbb{Q}$$ and $$E = \mathbb{R}$$ and finding all $$\alpha \in \mathbb{R}$$ such that
$$\phi_\alpha ( x^2 + x - 6 ) = 0
$$
that is finding all zeros of $$x^2 + x - 6$$ in $$\mathbb{R}$$ ... ... "My question is as follows:
What is the relevance of the field $$F$$? It appears that if we made $$F = \mathbb{R}$$ we would have achieved the same result ... .. indeed (if we regard a field as a subfield of itself) we could have taken $$F = E$$ and achieved the same result ...
Can someone please explain the relevance of the subfield $$F$$? ... ... I am sure that I am missing something ...
Peter
*** NOTE ***I do understand that changing $$F$$ changes the nature/type of the polynomials that can be input to the homomorphism $$\phi_\alpha$$ since the co-efficients of the polynomial come from $$F$$ ... but still do not really see the point or relevance of the subfield $$F$$ ...
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 $$r^2 + r - 6 = 0$$ by letting $$F = \mathbb{Q}$$ and $$E = \mathbb{R}$$ and finding all $$\alpha \in \mathbb{R}$$ such that
$$\phi_\alpha ( x^2 + x - 6 ) = 0
$$
that is finding all zeros of $$x^2 + x - 6$$ in $$\mathbb{R}$$ ... ... "My question is as follows:
What is the relevance of the field $$F$$? It appears that if we made $$F = \mathbb{R}$$ we would have achieved the same result ... .. indeed (if we regard a field as a subfield of itself) we could have taken $$F = E$$ and achieved the same result ...
Can someone please explain the relevance of the subfield $$F$$? ... ... I am sure that I am missing something ...
Peter
*** NOTE ***I do understand that changing $$F$$ changes the nature/type of the polynomials that can be input to the homomorphism $$\phi_\alpha$$ since the co-efficients of the polynomial come from $$F$$ ... but still do not really see the point or relevance of the subfield $$F$$ ...
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