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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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Proving the following Lie algebra only has inner derivations?

Let $K$ be the field with 2 elements and let $\mathfrak{g}$ be the Lie algebra over $K$ with a basis $\{a,b,c,d,e,f,g,h\}$ and the following multiplication table: \begin{array}{c|cccccccc} [\...
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2answers
64 views

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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26 views

Is this unipotent group, over characteristic 2, connected?

Let $E_{ij}(x)\in Mat_{7\times7}(\overline{\mathbb{F}}_2)$ be the matrix with zeros everywhere, except for the value $x$ at $ij$. Set $a(x)=1+E_{12}(x)+E_{34}(x)+E_{56}(x)$, $b(y)=1+E_{23}(y)+E_{45}(y)...
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1answer
47 views

Transforming Quadrics in Characteristic 2

I’m trying to solve the following problem given in a textbook: Let $k$ be an algebraically closed field and $Q=V(F)$ a quadric in $\mathbb{P}^3(k)$, where $F$ is an irreducible polynomial in $X,Y,Z,...
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1answer
19 views

A homomorphism on a nontrivial commutative ring with trivial unit group

Suppose $R \neq \{0\}$ is commutative and satisfies $R^\times = \{1\}$. I've shown that this implies that $\operatorname{char}R=2$ (by showing that $-1 = 1$). Now consider the homomorphism $f \in \...
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1answer
48 views

$x\mapsto x^p$ 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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1answer
150 views

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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20 views

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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43 views

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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1answer
57 views

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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1answer
27 views

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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1answer
41 views

“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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1answer
95 views

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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1answer
69 views

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 ...
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2answers
198 views

Peirce decomposition of a ring: must the ideal generators be idempotent in characteristic 2?

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 $J$, ...
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18 views

Transcendental variable is not square in $k(t)$

Let $k$ be a field of characteristic 2, and let $t$ be a transcendental variable over $k$. I want to conclude that there is no solution to $x^2 = t$ with $x \in k(t)$. I know that there is no ...
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0answers
22 views

Galois extension of exponent $mp^r$ in characteristic $p$

Kummer theory treats Galois extensions of exponents that are not divisible by the characteristic. Artin-Schreier and Witt extend this theory for Galois extensions of exponents $p^r$ in characteristic $...
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111 views

Characteristic of residue field in a Dedekind domain.

Let's consider a Dedekind domain $A$ with field of fractions $K$. Let $L$ be a finite Galois extension of $K$ and $B$ the integral closure of $A$ in $L$. Let $\mathfrak{p}\subset A$ be a non-zero ...
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1answer
63 views

A field with an irreducible, separable polynomial with roots $\alpha$ and $\alpha + 1$ must have positive characteristic.

Given a field $\mathbb{F}$ with an irreducible, separable polynomial $f(x),$ let $E$ denote the splitting field of $f$ over $\mathbb{F},$ and assume that $\alpha$ and $\alpha + 1$ are roots of $f(x).$ ...
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2answers
29 views

Characteristic of a finite field - equivalent formula

Let $\mathbb{K}$ be a field with characteristic $\text{char}(\mathbb{K}) = p$ where $p$ is a prime. It means that the sum of $p$ unities ($1+ 1+\ldots + 1$ added $p$ times) equals $0$. How to show ...
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1answer
129 views

Cyclic Artin-Scheier-Witt extension of order $p^2$

Let $k = \mathbb{F}_{p^r}(t)$. Artin-Schrier polynomials $f(x) = x^p - x - a \in k[X], a \in k$ describe all the cyclic Galois extensions $K/k$ of order $p$. To generalize to cyclic extensions of ...
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2answers
89 views

Characteristic of an integral domain: doubt in the proof

Suppose by absurd that the characteristic of an integral domain is an integer $n$ not prime, say $n=n_1 \cdot n_2$. Now we have $$na=(n_1 \cdot n_2)a=0 \implies (n_1 \cdot a) \cdot (n_2 \cdot a)=0 \...
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0answers
37 views

Weil pairing on $E[p]$ is trivial

I'm currently working through Elliptic Curves, L. C. Washington. On page 147 he writes "The Weil pairing is not defined on $E[p]$ (or, if we defined it, it would be trivial since $E[p]$ is cyclic and ...
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1answer
38 views

Multiple roots for roots of unity in fields of some characteristic

I'm currently working through Elliptic Curves: Number Theory and Cryptography, L. C. Washington. The bottom of page 82 says "Since the characteristic of $K$ does not divide $n$, the equation $x^n = 1$...
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4answers
706 views

Can a finite ring have a characteristic $0$?

I am working on studying Abstract Algebra and doing one of the review problems for the exam. I do not think a finite ring can have characteristic zero because when I think of, for example, Integers ...
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1answer
70 views

Middle Terms do not cancel correctly when calculating the Discriminant of the Frey Curve

It's in several references but I will use Hardy and Wright 6th edition p.587 where they use the generalized Weierstrass discriminant on p.558 to calculate the discriminant for the Frey Curve. They ...
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0answers
61 views

Finite generation of a pre-image of a finitely generated subalgebra

Let $k$ be a field, let $A$ and $B$ be commutative, finitely-generated, graded $k$-algebras, and let $S$ be a finitely-generated, graded $k$-subalgebra of $B$. Question1: If $\varphi:A\to B$ is a ...
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1answer
89 views

Additive subgroup of a field of characteristic $p$ is an elementary abelian $p$-group

In this paper (in the abstract), it is mentioned that: A finite subgroup of the additive group of a field $F$ of characteristic $p \neq 0$ is an elementary abelian $p$-group. Why is this so?
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3answers
161 views

$R$ a ring with 1 and charac. $n>1$ (resp. 0) $\implies R$ contains a subring $\simeq \Bbb{Z}_n$ (resp. $\Bbb{Z}$). What happens if $R$ has no 1?!

Definition: Let $R$ be a ring. We say that $n\in \Bbb{Z}_+$ is the characteristic of $R$ if it is the least positive integer such that $n r=0$, for all $r\in R$ (here $nr$ denotes $r+r+\dots+r$, "$n$ ...
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1answer
107 views

Discriminant in char $2$

Let $n\geq 1$, $$D=\prod_{i<j}(x_i-x_j)^2, B=\prod_{i<j}(x_i+x_j), C=(B^2-D)/4$$ be polynomials in $\mathbb{Z}[x_1, ..., x_n]$ Then if $f(x)\in F[x]$ has different roots $\alpha_1, ... \alpha_n$...
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3answers
76 views

Is the Polynomial $X^{p^n}-X$ the zero polynomial in characteristic $p$?

Suppose that $p$ is a prime number and that $\mathbb{F}_{p^n}[X]$ is the ring of polynomials with one variable, with coefficients in the field $\mathbb{F}_{p^n}$ for some $n\in\mathbb{N}$. Then $p(...
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0answers
27 views

$(k^s)^p$ generates $k^s$ over $k$ if $k$ is $F$-finite?

I am puzzled with the following (apparently easy ?) question. Let $k$ be a field of characteristic $p>0$, such that $[k:k^p]<\infty$ (in other words, $k$ is $F$-finite, with $F: k\rightarrow k$ ...
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2answers
154 views

Regular rings with F-finite field of fractions

Let $S$ be a regular domain of characteristic $p>0$ with fraction field $K$. Assume that $K$ is $F$-finite, meaning that $K$ is a finite module over $K^p$. Does it follow that $S$ is also $F$-...
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0answers
76 views

A statement about normal basis element , trace and characteristic of field

Let $F/K$ be a normal finite extension where $F=F_1 \times F_2$ for subfields $F_1,F_2$ of $F$ where $F_1\ne F$ . Suppose $w$ is a normal basis element in $F/K$ for any $w\in F$ for which $Tr_{F/F_2}(...
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1answer
83 views

$x^2 + x$ in a finite local ring of even characteristic and its residue field

Suppose $R$ is a finite commutative local ring of characteristic $2^n$ with unique maximal $M$ and its residue field $k$ (of characteristic $2$). Consider the canonical map $\pi : R \rightarrow k$ by $...
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1answer
281 views

$R$ be a commutative ring with unity of prime characteristic $p$ , if $a\in R$ is nilpotent then is $1+a$ unipotent?

Let $R$ be a commutative ring with unity of prime characteristic $p$ , if $a\in R$ is nilpotent then does $\exists n \in \mathbb N$ such that $(1+a)^n=1$ ?
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1answer
87 views

What is the Frobenius automorphism associated to a closed pont?

Let X be a variety over a finite field. Let $x$ be a closed point of $X$. Then from it we are supposed to get an automorphism $Frob_x$ that induces a map from the group of zero cycles to the ...
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1answer
137 views

About a refined Burnside's theorem in prime characteristic.

The following result is due to W. Burnside: Theorem. Let $G$ be a subgroup of $\textrm{GL}_n(\mathbb{C})$. If $G$ has finite exponent, then $G$ is finite. The proof relies on the following: ...
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1answer
343 views

Determinant of Killing form of sl_n

My question is something like a continuation of: Is there a more intelligent way to compute the determinant of the Killing form of $\mathfrak{sl}(3,F)$? Motivated by that same problem in Humphreys' ...
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0answers
110 views

What goes wrong in the theory of algebraic groups in characteristic $p$?

I wonder about the basic facts in the theory of algebraic groups and Lie algebras that are wrong in characteristic $p$. For example, The trace of $1$, that is the dimension of a representation, may ...
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1answer
501 views

Does there exist a pair of infinite fields, the additive group of one isomorphic to the multiplicative group of the other?

It is a common exercise in algebra to show that there does not exist a field $F$ such that its additive group $F^+$ and multiplicative group $F^*$ are isomorphic. See e.g. this question. One of the ...
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1answer
200 views

Determine all $P(X)\in K[X]$ such that $P\big(X^2+1\big)=\big(P(X)\big)^2+1$, for fields $K$ of any characteristic.

This question is inspired by this thread. However, in this question, I take an arbitrary field instead of $\mathbb{R}$ and drop the assumption that $P(0)$ must be $0$. Let $K$ be a field. ...
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1answer
368 views

Canonical connection on the Frobenius pull-back

If $X$ is a scheme over a scheme $S$ of characteristic $p>0$ and $F:X\to X^{(p)}$ is the relative Frobenius, it is known that there is a canonical connection on the Frobenius pull-back $F^*E$ of a ...
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1answer
60 views

Does $2x=0$ imply $x=0$ in a field with characteristic not equal 2.

Let $F$ be a field with $char(F)\neq 2$. If $2x=0$ with $x\in F$, can we get $x=0$?
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2answers
235 views

Let $R$ be a commutative unital ring. Is it true that the group of units of $R$ is not isomorphic with the additive group of $R$?

Let $R$ be a commutative ring with unity, and let $R^{\times}$ be the group of units of $R$. Then is it true that $(R,+)$ and $(R^{\times},\ \cdot)$ are not isomorphic as groups ? I know that the ...
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1answer
154 views

Example of projection map having non-reduced fibers

This question stems from Oliver Debarre's Higher-Dimensional Algebraic Geometry, proposition 5.7. Let $X$ be a normal quasi-projective variety over an algebraically closed field of characteristic $p &...
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0answers
20 views

Orders of elements in multiplicative groups of fields with positive characteristic

Suppose that $k$ is a field with positive characteristic $p$. I want to show that for each $x\in k$ the set $\{x^n\colon n\in \mathbb{N}\}$ is finite. My intuition tells me that I should use Fermat's ...
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1answer
280 views

Idempotents in commutative ring of characteristic 2 form a subring

In a commutative ring of characteristic $2$, I want to show that the idempotents form a subring. It's probably easiest to directly test that the set of forms a subring. It is easy to verify ...
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0answers
297 views

Kähler differentials in an inseparable field extension

Let $L/K$ be a finite (or, more generally, algebraic) field extension. It is easy to show that if $L/K$ is separable then the $L$-vector space $\Omega_{L/K}$ of relative Kähler differentials is zero. ...
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1answer
379 views

Can a separable isogeny of elliptic curves have an inseparable dual?

Let $\phi: E_1\to E_2$ be an isogeny of elliptic curves over a field $K$ of characteristic $p>0$. Suppose that $\phi$ is separable and let $\hat{\phi}: E_2\to E_1$ denote the dual isogeny. Then $\...