In the polynomial ring section of the textbook that I am reading, it states:
$F_2[x]/(x^2 + x) ≅ F_2[x]/(x) × F_2[x]/(x + 1) ≅ F_2 × F_2$.
I know how to show the first pair of isomorphism. Since $(x, x+1) = (1)$, we can use the Chinese Remainder Theorem to show that $F_2[x]/(x^2 + x) ≅ F_2[x]/(x) × F_2[x]/(x + 1)$.
But how do we show that $F_2[x]/(x) × F_2[x]/(x + 1) ≅ F_2 × F_2$?
Thank you!