Questions tagged [positive-characteristic]

For questions involving rings of positive characteristic, particularly those questions whose statements and hypotheses heavily rely on positive characteristic. Consider using the ring-theory tag for questions that apply to rings of all characteristics.

In the category of rings with unity and unity-preserving ring homomorphisms, the integers are an initial object. This means for each ring $R$, there is a unique map $\mathbb{Z}\to R$. If the map is injective, we say $R$ has characteristic $0$. If this map is not injective, its kernel must have the form $n\mathbb{Z}$ for some natural number $n$. In this case, we say that the characteristic of $R$ is $n$, and we write $\operatorname{char}R=n$. Typical examples of such rings are the finite fields along with their extensions. This tag is for questions involving rings such that $\operatorname{char}R=n>0$.

Topics such as representation theory and algebraic geometry have very different tastes depending on the characteristic of the base field $k$. For example, certain foundational theorems in characteristic $0$ are either open questions in positive characteristic, or known to be false. See resolution of singularities and Maschke's theorem.

188 questions
5
votes
2 answers

infinite ring with non-zero finite characteristic

For every natural number n, i have to give a finite and a infinite ring with characteristic n. For the finite case is it simple: for all n $\in\mathbb{N}$, the ring $\mathbb{Z}_n$ is a finite ring with characteristic n. But in the infinite case i…
2
votes
3 answers

the positive square root of $I$?

Find operators $T \colon \mathbb{R}^2 \to \mathbb{R}^2$ such that $T^2=I$. Which one is the positive square root of $I$? Is there the operator $T$ such that $T(T(x))=I$ where $I(x)=x$?