= Spectrum of a finite-dimensional commutative algebra
{title2=$\operatorname{Spec}C=\operatorname{MaxSpec}C,\quad|\operatorname{Spec}C|\leq\dim_KC$}
If $C$ is a commutative algebra of finite dimension over a <field> $K$, every <prime ideal> is maximal and $|\operatorname{Spec}C|\leq\dim_KC$. Indeed, the quotient by a prime is a finite-dimensional <integral domain>. Multiplication by any nonzero element is an <injective> linear <endomorphism>, hence <surjective>, so the quotient is a <field>. Distinct <maximal ideals> are pairwise comaximal. For any finite list $\mathfrak m_1,\ldots,\mathfrak m_r$, construct $e_i\equiv1\pmod{\mathfrak m_i}$ and $e_i\equiv0\pmod{\mathfrak m_j}$ for $j\ne i$ by multiplying elements in $\mathfrak m_j$ congruent to one modulo $\mathfrak m_i$. These give a <surjection> $C\to\prod_iC/\mathfrak m_i$. Each factor has positive $K$-dimension, so $r\leq\dim_KC$, excluding an infinite list.
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