# 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.

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### Monogenic function fields

Recall that a number field $K$ is said to be monogenic if its ring of integers is of the form $\mathbb{Z}[\alpha]$, or equivalently, if it has a power integral basis. There are many references one can ...
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### Example of ring can’t be defined over finite field.

Let $K$ be an infinite field of characteristic $p>0$, and $A$ be a $K$-algebra. We say that $A$ can be defined over a finite field if there is finite subfield $F\subset K$ and an $F$-algebra $A_0$ ...
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### Regularity vs smoothness in positive characteristic

It is well known that a scheme over a perfect field is smooth at $x$ if and only if it is regular at $x$, and that these two properties are not equivalent over non-perfect fields. What is an example ...
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### Algebraic Peter-Weyl in positive characteristic

To my understanding there is an algebraic version of Peter-Weyl that holds in characteristic $0$ that says for any reductive group $G$ one has that: $$k[G]=\bigoplus V\otimes V^*$$ as a $G\times G$-...
1 vote
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### Proof verification of the theorem that integral domains have either $0$ or prime characteristic

I am doing Contemporary Abstract Algebra by Gallian and am stuck at the proof of the fact that integral domains have either $0$ or prime characteristic. The proof in the book goes as follows: Let $R$ ...
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### General position theorem for curves: do we use characteristic zero in this proof?

I'm reading about the general position theorem in Arabello, Cornalba, Griffiths, and Harris' Geometry of Algebraic Curves volume 1. Everything is over $\Bbb C$, and for these authors a curve is a ...
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### Prove an element not belonging to the tight closure of an ideal

I'm working on a ring $R=\mathbb{F}_7[x,y,z]/(x^2+y^3+z^5)$ and want to prove $x\notin(y,z)^*$, the tight closure. First I want to find a test element, which can be obtained from the Jacobian ideal. ...
1 vote
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### Surjective proper cover by ordinary varieties

Let $X$ be smooth proper variety over a finite field $k$ of positive characteristic $p$. Assume that $X$ is not ordinary, then my question is if there exists a smooth, projective ordinary $Y$ with a ...
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### Why is the Levi-decomposition not available in positive characteristic?

I am learning about Lie-Algebras and I came across a proof of the Levi-decomposition. Can somebody tell why the Lie-Algebra has to be over a field of characteristic $0$? Why does it fail for positive ...
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1 vote
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### Lie groups over fields of finite characteristic

Does anyone have any good references on Lie groups over fields of finite characteristic? I am trying to find something comprehensive that shows what fails and what succeeds in comparison to Lie theory ...
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### A property of fields of characteristic $p$ and their extensions

I am currently stuck on the following problem, taken from Herstein's Topics in Algebra, 2nd ed.. It reads: If $F$ is of characteristic $p \neq 0$ and if $K$ is a finite extension of $F$, prove that ...
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### Algebraic Curves in Positive Characteristic Reference

I am looking for an (advanced) book on algebraic curves that treats positive characteristic, something beyond Hartshorne Chapter IV. I am looking for material close to Geometry of Algebraic Curves by ...
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### Fields of prime characteristic in which an irreducible polynomial has a root of multiplicity > 1

In the following solution to ex21 (iii) I do not understand the second last line. Is there a mistake? The first summation implies the degree of $f$ is $n$ whereas the second summation implies the ...
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### Is a normal irreducible separable polynomial of degree $p$ over a field of characteristic $p$ necessarily Artin-Schreier

Is a normal irreducible separable polynomial of degree $p$ over a field of characteristic $p$ necessarily Artin-Schreier? If it is true, I am also wondering whether we can generalize this in the ...
Suppose $f:X \to Y$ is a finite morphism, lets say of surfaces for simplicity. If $|A|$ is a very ample linear system on $X$ then any general $H \in |A|$ is smooth, in any characteristic, assuming ...
Let $X$ be a smooth affine scheme over $\mathbb{Z}/{p^2}$ that is a complete intersection, say $X$ is the spectrum of $\mathbb{Z}/{p^2}[x_1,...x_n]/(f_1, ... f_r)$, where $n-r$ is the dimension of $X$....