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I am reading Dummit and Foote, Chapter 13 - Field Theory.
I am currently studying Theorem 3 [pages 512 - 513]
I need some help with an aspect of the proof of Theorem 3 concerning congruence or residue classes of polynomials.
D&F, Chapter 13, Theorem 3 and its proof read as follows:https://www.physicsforums.com/attachments/2712
View attachment 2713In the above text D&F state the following:
" ... ... If $$ \overline{x} = \pi (x) $$ denotes the image of x in the quotient K, then
$$ p ( \overline{x} ) = \overline{p(x)} $$ ... ... ... (since $$ \pi $$ is a homomorphism) ... ... "
I do not completely understand this statement and further I am having some trouble interpreting the meaning of the notation ...
I hope someone can help .. ...
In order to ensure I understood congruence classes or residue classes in F[x] I went to Hungerford, Abstract Algebra: An Introduction, Chapter 5. In this chapter Hungerford gives the following definitions:View attachment 2714
View attachment 2715So, following Hungerford, if we want to find the residue class of $$ f(x) = x $$, we write:
$$ \overline{x} = [x] = \{ x + k(x)p(x) \ | \ k(x) \in F[x] \} $$
But given this ... how do we form $$ p( \overline{x})$$?
Do we substitute $$p(x)$$ for $$x$$, everywhere $$x$$ appears getting the following:
$$ p ( \overline{x} ) = p[x] = \{ p(x) + k(p(x))p(p(x)) \ | \ k(x) \in F[x] \} $$ ?
I know this seems clumsy ... but it seems to me to have some formal merit ... anyway, although I suspect it is not the way to go ... I am not sure why ...
So maybe the right interpretation is as follows:
$$ p ( \overline{x} ) = p[x] = \{ p(x) + k(x)p(x) \ | \ k(x) \in F[x] \} $$ ?
Can someone clarify this issue for me indicating not only which alternative is correct - but further why that alternative is correct and the other wrong.
I would appreciate some help in this matter.
Peter
I am currently studying Theorem 3 [pages 512 - 513]
I need some help with an aspect of the proof of Theorem 3 concerning congruence or residue classes of polynomials.
D&F, Chapter 13, Theorem 3 and its proof read as follows:https://www.physicsforums.com/attachments/2712
View attachment 2713In the above text D&F state the following:
" ... ... If $$ \overline{x} = \pi (x) $$ denotes the image of x in the quotient K, then
$$ p ( \overline{x} ) = \overline{p(x)} $$ ... ... ... (since $$ \pi $$ is a homomorphism) ... ... "
I do not completely understand this statement and further I am having some trouble interpreting the meaning of the notation ...
I hope someone can help .. ...
In order to ensure I understood congruence classes or residue classes in F[x] I went to Hungerford, Abstract Algebra: An Introduction, Chapter 5. In this chapter Hungerford gives the following definitions:View attachment 2714
View attachment 2715So, following Hungerford, if we want to find the residue class of $$ f(x) = x $$, we write:
$$ \overline{x} = [x] = \{ x + k(x)p(x) \ | \ k(x) \in F[x] \} $$
But given this ... how do we form $$ p( \overline{x})$$?
Do we substitute $$p(x)$$ for $$x$$, everywhere $$x$$ appears getting the following:
$$ p ( \overline{x} ) = p[x] = \{ p(x) + k(p(x))p(p(x)) \ | \ k(x) \in F[x] \} $$ ?
I know this seems clumsy ... but it seems to me to have some formal merit ... anyway, although I suspect it is not the way to go ... I am not sure why ...
So maybe the right interpretation is as follows:
$$ p ( \overline{x} ) = p[x] = \{ p(x) + k(x)p(x) \ | \ k(x) \in F[x] \} $$ ?
Can someone clarify this issue for me indicating not only which alternative is correct - but further why that alternative is correct and the other wrong.
I would appreciate some help in this matter.
Peter