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How do we compute the dimension as a $k$-vector space $(\mathrm{char}(k) \neq 2)$ of $k[x,z]/(x^{2}+1,z^{2})$?

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    Aren't $1,x,z,xz$ linearly independent?2012-05-22

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Succinctly: $k[x,z]/(x^2 + 1, z^2) \simeq k[x]/(x^2 + 1) \otimes_k k[z]/z^2$Morally, it's cuz $x$ and $z$ have nothing to do with each other. Formally, these bad boys are isomorphic as $k$ algebras, hence rings, since to map out of either just requires choosing elements of the target $x', z'$ satisfying $x'^2 = -1, z'^2 = 0$.

So the dimension of the whole thing is the product of their individual dimensions, and $k[t]/p(t)$ has dimension degree of $p$, so 4.