I am going through a proof of a question that asks:
Prove $\langle$x+1,x$^2$+1$\rangle$ is a maximal ideal in $\Bbb Z$[X]
The proof shows that this ideal is equal to $\langle$x+1,2$\rangle$ and then uses the following;
"Every polynomial f(x)$\in$ $\Bbb Z$ can be expressed by f(x)=(x+1)g(x) + f(-1)"
How would I go about proving this equation?
I can see this comes from Euclids Algorithm with f(x)=q(x)g(x)+r(x) with deg(r) < deg(q) = 1, so r is a constant, and I can see how this works, but am not sure how to conclude that r must be equal to f(-1)
For f a constant and when (x+1) is a factor of f this is obvious but in the final case, (non-constant, (x+1) not a factor) i am unsure how to show generally
I'm sure I'm missing something very obvious, any hints would be appreciated.
Thanks.
Since you know from Euclid's algorithm that $$ f(x)=(x+1)g(x)+r $$ where $r$ is a constant, you just plug in $x=-1$ and you find that $f(-1)=r$. This is called "the Remainder's Theorem".