Prove that certain elements are not in some ideal I have the following question:
Is there a simple way to prove that
$x+1 \notin \langle2, x^2+1\rangle_{\mathbb{Z}[x]}$ and
$x-1 \notin \langle2, x^2+1\rangle_{\mathbb{Z}[x]}$
without using the fact that $\mathbb{Z}[x]/\langle2, x^2+1\rangle$ is an integral domain?
Thanks for the help.

Perhaps the simplest way to try to address these questions is tro try to characterize the elements in the ideal. In this case
$$f(x)\in\langle 2,x^2+1\rangle\iff f(x)=2g(x)+(x^2+1)h(x)\;,\;\;g(x),h(x)\in\Bbb Z[x]$$
and this means that any non-zero element in the ideal has either all its coefficients even and/or it is a multiple of $\,x^2+1\,$ , and this means either $\,f\,$ has all its coefficients even or its degree is at least two
Well, since neither $\,x-1\,,\,x+1\,$ fulfills the above neither belongs to the ideal.
