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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1answer
31 views

Root of $f$ is $p^{\text{th}}$ power in extension field $\Rightarrow$ coefficients of $f$ are $p^{\text{th}}$ powers in base field.

Let $K$ be a field of characteristic $p > 0$, and $f ∈ K[X]$ monic and irreducible with root $\alpha$. Let $F$ be the Frobenius endomorphism. To demonstrate: $\alpha ∈ F[K(\alpha)] \Rightarrow f ∈...
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1answer
37 views

$X^{p^k} - a ∈ K[X]$ irreducible?

Let $K$ be a field of characteristic char$(K) = p > 0$, and let $a ∈ K$ be an element with the following property: $$(\forall \beta ∈ K)(\beta^p ≠ a). $$ Let $k ∈ ℕ$ be arbitrarily given. Is it ...
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23 views

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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42 views

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 ...
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27 views

Bertini for finite morphisms

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 ...
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40 views

Lifting a Frobenius endomorphism under an étale morphism

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 $X$....
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31 views

Closed form expression for Taylor-like series in 2 variables in positive characteristic

Given a polynomial function $f$ on $1$ variable, the following Taylor series: $$\sum_{n=0}^\infty \frac{h^n}{n!} \frac{d^n f}{dx^n}(x)$$ can be written in closed form as $f(x+h)$. (A cool fact to ...
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47 views

Characteristic of coefficient field of elliptic curve

Characteristic_(algebra) In mathematics, the characteristic of a ring $R$, often denoted $\operatorname{char}(R)$, is defined to be the smallest number of times one must use the ring's multiplicative ...
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35 views

Frobenius pullback of $\mathcal O_X$ and Frobenius twist

Let $k$ be a perfect field of characteristic $p > 0$ and let $X \to S = \textrm{Spec } k$ be a scheme. Let $F : X \to X$ be its absolute Frobenius and let $X' := X \times_{F_S} S$ be the Frobenius ...
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41 views

restrictions to the element $b$, such that $b=\sigma (a)$ where $\sigma$ is a generator of the Galois group

Let $K/L$ be a finite Galois extension of degree $p$, $p$ is prime ($char \: L =p$, $p$ is prime). As I know $K=L(a)$ for some $a\in L$. Is there a way to find any restrictions to the element $b$, ...
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1answer
48 views

Field extension of a field of Characteristic $p$ [closed]

Let $K$ be extension of $F$ with characteristic of $F$ being $p$ and $f(x)$ a monic polynomial in $K[x]$ with $[f(x)]^m$ belongs to $F[x]$ such that $p$ does not divides $m$. Then show that $f(x)$ ...
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1answer
33 views

Prime factors of a separable polynomial in a non-splitting field.

Let $K$ be a field of characteristic $p$ and consider a polynomial $p(x) = x^n - a \in K[x]$, with $a \neq 0$ and $p \nmid n$. By the derivative test this is certainly separable. Even if $p$ does ...
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1answer
42 views

Semisimple representation theory of finite groups in characteristic p.

For $G$ a finite group, and algebraically closed field $k$ of finite characteristic coprime to $|G|$, we have that $k[G]$ is a semisimple algebra. This gives complete reducibility, and I vaguely ...
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1answer
115 views

A corollary of Schur's lemma in positive characteristic

A corollary of Schur's lemma reads (nLab): In the case that the ground field is an algebraically closed field of characteristic zero; endomorphisms $\psi:V \rightarrow V$ of a finite dimensional ...
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1answer
68 views

Do invariants and co-invariants for a cyclic group have the same dimension?

Let $V$ be a finite dimensional vector space over $\mathbb{F}_p$ with an action of the cyclic group $C_p$ of order $p$. We can now form the two vector spaces of invariants $V^{C_p}$ and co-invariants ...
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44 views

Algebraic de Rham cohomology of the affine plane in characteristic p?

The fact that $H^1_{\mathrm{dR}}(\mathbb{A}^1_k) = \bigoplus_{i=0}^{\infty}k$ for $k$ of positive characteristic exemplifies what can go wrong with de Rham cohomology when the characteristic is not ...
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1answer
20 views

Character group of diagonalizable group has no $p$-torsion

Let $G$ be a diagonalizable algebraic group over a field $K$ of characteristic $p > 0$. Let $X$ be the character group of $G$ (algebraic group homomorphisms $G \to K^\times$). We know $X$ is ...
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122 views

Frobenius twist of a field

Let $k$ be a field of characteristic $p>0$ (not necessarily perfect). Consider the Frobenius endomorphism $F : k \to k$, $x \mapsto x^p$. I am curious about what happens when we take $k$ as a $k$-...
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143 views

The group algebra is not semisimple if characteristic divides group order.

I'm studying a proof that if a prime $p$ has $p\mid |G|$ and $k$ is a field of characteristic $p$, then the group algebra $kG$ is not semisimple. My issue is that there is an assertion in the first ...
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59 views

A question on regular local rings (of positive characteristic ) of dimension $2$

Let $R$ be a regular local ring of dimension $2$ and of characteristic $p>0$. How to show that for every $f_1,f_2,f_3 \in R$, $\exists 0\ne c\in R$ and $n_0\in \mathbb N$ such that $c(f_1f_2f_3)^{...
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64 views

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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109 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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1answer
68 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
34 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
118 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
60 views

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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1answer
158 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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62 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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48 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
66 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
67 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
85 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
135 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
153 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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263 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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42 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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195 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
72 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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46 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
339 views

Cyclic Artin-Schreier-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
114 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
70 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
67 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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5answers
1k 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
75 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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75 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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406 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
156 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
112 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(...