Verification of quotient ring by Chinese remainder theorem

Question

If F is a field then is $F[x]/\left<x^2\right> = F \times F$ correct by CRT ? Can we write $F[x]/\left<x^2-1\right> = F \times F$ by CRT ? I think both can be written in the corresponding forms. If any mistake, please verify it.

Answer 1

The first guess you gave will always be wrong. The quotient by $x^2$ has nonzero nilpotents but $F\times F$ does not. It cannot be “correct by the CRT” because the CRT doesn’t apply. I think you need to go back and read it again.

If $F$ has characteristic $2$ the second guess also fails for the same reason.

For fields not of characteristic $2$ though the Chinese remainder theorem will take hold because $R=(2)=(x+1, x-1)$, the factors are coprime.