# 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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### If the characteristic of a field is two, doesn't the quadratic form associated with a bilinear form exist?

I have a question: if the characteristic of a field is two, doesn't the quadratic form associated with a bilinear form exist?
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### Finite Characteristic of a Ring [duplicate]

If $H$ is a ring and it has a finite non-zero characteristic $p$ then ring is finite. I couldn’t any counter example for this claim. Can anyone help me please?
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### Are simple Lie algebras complete?

Let $\mathfrak{g}$ be a finite dimensional simple Lie algebra over a field of characteristic $p >0$. Is $\mathfrak{g}$ complete? If not, under what conditions is $\mathfrak{g}$ complete? A Lie ...
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### $x\mapsto x^p$ with $\operatorname{char}(F)=p>0$ is a ring homomorphism [closed]

Let a field $F$ with $\operatorname{char}(F)=p>0$. Let a map $f:F\to F$ defined be $x\mapsto x^p$. Show that $f$ is a ring homomorphism. Obviousely $0\mapsto 0$ and $(xy)^p=x^py^p$. How can I show ...
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### Why does the cardinality of the vector space over a finite field of characteristic $p$ have to be a power of $p$?

In a lecture note that I have, it is written that if $F$ is a field of $q$ elements of characteristic $p$, then $q = p^m$ for some $m>0$. To show this, observe that $F$ is a vector space ...
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### Lang Steinberg over separably closed field

Let $K=K^{sep}$ be a separably closed field with $K|\mathbb{F}_q$, where $\mathbb{F}_q$ is the field with $q$ elements. Let $\mathbb{G}$ be a connected linear algebraic group over $\mathbb{F}_q$. ...
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### Stability under conjugation of a subvector space of Lie algebra of an affine linear algebraic group (in positive characteristic)

Let $G$ a matrix Lie group and $\mathfrak{g}$ its Lie algebra. If a subvector space $V$ of a Lie algebra $\mathfrak{g}$ is stable under conjugation, then it's an ideal. I know this is should be true ...
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### Reason why $x \mapsto x^p$ is a ring morphism in characteristic $p$?

Let $R$ be a commutative unital ring of characteristic $p$, where $p$ is prime. The Frobenius map $x \mapsto x^p$ on $R$ is known to be a ring homomorphism (in particular, additive). The only proof I ...
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### Rings in which each element is a sum of $n$ commuting idempotents

Let $n$ be a nonnegative integer. Let $R$ be a nonunital ring such that every element of $R$ is a sum of $n$ pairwise commuting idempotents. (As usual, the class of nonunital rings includes the class ...
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### Abelian group of prime exponent $p$ is $\mathbb{F}_p$-vector space?

In reading about group cohomological, I came across the following in a statement of a lemma. "Let $p$ be a prime, and let $A$ be an abelian group of exponent dividing $p$..." Is this just a ...
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### “Purely inseparable, algebraic extension if and only if trivial automorphism group”

Let $k \subset K$ be an algebraic field extension, with $\mathrm{char} \, k=p>0$. Is it true the following assertion? $k \subset K$ is purely inseparable if and only if $\mathrm{Aut}_k(K)=\{1\}$; ...
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### Example of a characteristic zero local ring with a quotient of positive characteristic

This question was featured on a qualifying exam at my university: What's an example of a commutative local ring $R$ of characteristic zero, with a non-maximal prime ideal $P$ such that the ...
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### Indecomposable modules of a $p$-group in characteristic $p$

Let $k$ be a field of characteristic $p>0$ and let $P$ be a $p$-group. Then the group ring $kP$ has only one simple module, the trivial module $k$, and hence there is only one projective ...
I'm considering this particular statement of the Peirce decomposition of a ring: If a commutative unital ring $R$ can be written as an internal direct sum of two of it's proper ideals $I$ and $J$, ...