= Mod-p cohomology ring of a cyclic group
{title2=$H^*(C_p;\mathbb F_p)$}
For a prime $p$, a classifying <Eilenberg–MacLane space> for $C_p$ is the infinite lens space $S^\infty/C_p$, using scalar multiplication by a primitive $p$th root on the complex-coordinate sphere. Its contractible <universal cover> is $S^\infty$. The standard two-periodic free resolution has differentials $g-1$ and $1+g+\cdots+g^{p-1}$, both zero after applying <cochains> with trivial $\mathbb F_p$ coefficients. Thus every <cohomology> degree has dimension one. If $p=2$, the ring is $\mathbb F_2[t]$, $|t|=1$. If $p$ is odd, it is $\Lambda(a)\otimes\mathbb F_p[b]$, with $|a|=1$, $|b|=2$: graded commutativity gives $a^2=0$ and the periodicity class $b$ induces the degree-two isomorphisms. The <Künneth theorem> gives two copies of these generators for $C_p\times C_p$; the dimension in degree $d$ is $d+1$ for every $p$.
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