(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let F be a field and f(x) in F[x]. If c in F and f(x+c) is irreducible, prove f(x) is irreducible in F[x]. (Hint: prove the contrapositive)

2. Relevant equations

So, im going to prove if f(x) is reducible then f(x+c) is reducible.

3. The attempt at a solution

f(x) = g(x)h(x) for g,h in F[x]. If [tex]f(x) = \sum{a_{i}x^i} = \sum{b_{i}x^i} \cdot \sum{c_{i}x^i} = g(x)h(x)[/tex] then [tex]f(x+c) = \sum{a_{i}(x+c)^i}[/tex]

Now i just actually work through the algebra and after all is said and done, i should see that its equivalent to

[tex]\sum{b_{i}(x+c)^i} \cdot \sum{c_{i}(x+c)^i}[/tex] ?

Now, if this is a correct approach, i was thinking it involves alot of work.

I was thinking about the evaluation homomorphism [tex]\phi_{x+c}:F[x] \rightarrow F

[/tex] given by (including notational convenience) [tex]\phi_{x+c}(f(x)) = \phi(f,x+c) =

f(x+c)[/tex] i.e. f evaluated at element x+c.

I know [tex]\phi[/tex] is a surjective homomorphism; so if f(x) is reducible,

[tex]\phi(f,x+c) = \phi(gh,x+c) = \phi(g,x+c)\phi(h,x+c)[/tex] but i cannot take that last part and equate it to [tex]g(x+c)h(x+c)[/tex] unless [tex]\phi[/tex] were injective.

Now, I know that in infinite field F, F[x] is isomorphic to the ring of its induced functions. That is to say, if F is infinite, then any two polynomials that look the same, act the same.

And vice versa, if two induced functions act the same, they are the same looking polynomial.

I am desiring some kind of isomorphism, call it [tex]\gamma[/tex] so i could simply

do this : [tex]\gamma(f, x+c) = \gamma(gh, x+c) = \gamma(g,x+c)\gamma(h,x+c) = g(x+c)h(x+c)[/tex] but i dont know how to get there.

Someone had suggested adapting evaluation homomorphism instead of from F[x] to F, i have it go F[x] to F[x] by treating the x as a polynomial instead of an element of F

such that [tex]\phi(f,x+c) = f(x+c)[/tex]; then showing if [tex]\phi(f(g),x)=\phi(f,\phi(g,x))[/tex] where that f(g) is formal composition and the right hand side is functional composition, it would be relevant.

Please, any thoughts? Very curious to know more of whats going on. I know there are many things going on here. Please help. THanks.

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# Homework Help: Abstract algebra: f(x) is reducible so is f(x+c)

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