- #1
VinnyCee
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Homework Statement
Let [tex]n\,\in\,\mathbb{N}[/tex]. Let F be a field, and suppose that [tex]p(x)\,\in\,F[x][/tex] is a polynomial of degree (n + 1).
Let S be the set:
[tex]S\,=\,\left\{\left(a_0,\,\ldots,\,a_n\right)\,:\,a_i\,\in\,F\right\}[/tex]
Define [tex]\phi[/tex]: [tex]S\,\rightarrow\,F[x]/\left(p(x)\right)[/tex] via
[tex]\phi\left(\left(a_0,\,\ldots,\,a_n\right)\right)\,=\,\left[a_0\,+\,a_1\,x\,+\,\cdots\,+\,a_n\,x^n\right][/tex]
Prove that [tex]\phi[/tex] is a bijection.
Homework Equations
[tex]f\,:\,B\,\rightarrow\,C[/tex] is injective provided that whenever f(a) = f(b) in C, then a = b in B.
[tex]f\,:\,B\,\rightarrow\,C[/tex] is surjective iff I am f = C.
[tex]f\,:\,B\,\rightarrow\,C[/tex] is bijective provided that f is both injective and surjective.
If [tex]f\,:\,B\,\rightarrow\,C[/tex] is a function, then the image of f is this subset of C:
[tex]Im\,f\,=\,\left{c\,|\,c\,=\,f(b)\,for\,some\,b\,\in\,B\right}\,=\,\left{f(b)\,|\,b\,\in\,B\right}[/tex]
The Attempt at a Solution
Prove Injectivity:
Here, I need to prove that whenever F[a]/(p(a)) = F/(p(b)) in F[x]/(p(x)), then a = b in S.
Now I need expressions for F[a]/(p(a))...
[tex]F[a]\,\equiv\,g(a)\,\left(mod\,p(a)\right)[/tex]
And F/(p(b))...
[tex]F\,\equiv\,h(b)\,\left(mod\,p(b)\right)[/tex]
Now how do I show that a = b in S?
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