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Questions tagged [positive-characteristic]

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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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K-isomorphic splitting fields in field with characteristic 2

The problem is as follows: Let $K$ be a field of characteristic $2$, let $a$ be an element of $K$ which is not of the form $b^2 + b$ for any $b \in K$, let $f_a(X) = X^2 + X + a$, and let $L_a = K(\...
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41 views

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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1answer
52 views

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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78 views

On finite separable morphisms

Let $f:X\longrightarrow Y$ be a finite separable morphism of smooth projective varieties over an algebraically closed field $k$. Let $d$ be the degree of $f$. We have an exact sequence $$0\...
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90 views

Cuspidal cubic in characteristic 3 with no inflection points?

Fix an algebraically closed field $k$ of characteristic $3$. Consider a curve in $\mathbb{P}^2_{k}$ given by $$xy^2+yz^2+zx^2=0.$$ It seems to be a cubic with a cusp at $[1:1:1]$, whose Hessian $(x+y+...
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36 views

Criteria for a morphism to be etale in positive characteristic

Are there any criteria for a morphism (as nice as you wish) in characteristic p to be etale that relates to the differential map on the associated sheaves of differentials or tangent spaces? I know ...
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29 views

Loday-Quillen-Tsygan Theorem

The Loday-Quillen-Tsygan states that for an associative algebra $A$ over a field of characteristic zero there is the isomorphism $H_* (gl(A)) \simeq \Lambda (HC_{*-1} (A))$. The proof heavily relies ...
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27 views

The equation $uX^p+X+d^p=0$ in $k((x))$

Let $k$ be a finite field in characteristic $p$ and let $K$ be the field $k((x))$. Let $u$ be a unit of $k[[x]]$ and let $d\in K$ but not in $k[[x]]$. Does the polynomial $uX^p+X+d^p$ have a roots in $...
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2answers
106 views

Let $\alpha$ be a root of $(x^2-a)$ and $\beta$ be a root of $(x^2-b)$. Provide conditions over $a$ and $b$ to have $F=K(\alpha+\beta)$.

QUESTION: Let $K$ be a field of characteristic different of 2. Let $F$ be a splitting field for $(x^2-a)(x^2-b)\in K[x]$. Let $\alpha$ be a root of $(x^2-a)$ and $\beta$ be a root of $(x^2-b)$. ...
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1answer
23 views

Simple extension of purely inseparable extension

It is Albert "Modern Higher Algebra", chapter 7, section 9, exercises 5 and 6. Let $K$ be a field of degree $n$ over $F$ of characteristic $p$ such that every quantity $\alpha$ of $K$ is a ...
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1answer
44 views

Are soluble/nilpotent lie algebras always isomorphic to a subalgebra of upper triangular matrices?

For soluble lie algebras, every representation has some basis such that the image is a subalgebra of upper triangular matrices (if you assume the field is algebraically closed). Then by Ado's theorem, ...
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28 views

Is there a ring $R$ such that $(R,+) \cong S^{1}$? [duplicate]

Here is the question I wanted to answer letter(c) of it: Let $R$ be a ring (with 1), and let $H$ be the cyclic subgroup of $(R,+)$ generated by $1.$ The characteristic of $R$ is $n$ if $H\cong \mathbb{...
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33 views

Squares on local rings of characteristic 2

Let $R$ be a local commutative ring with unit such that $2 = 0$ in $R$, that is, $\text{char}(R) = 2$. Is there an example of such a ring with the map $x \mapsto x^2$ not being surjective on the ...
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59 views

Characters of the nullcone in positive characteristic

In Jantzen's "Nilpotent Orbits in Representation Theory", in Section 8.18 (page 103), he derives (under certain conditions on the group $G$) a character formula for the coordinate ring $K[\...
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1answer
27 views

Prove that $m$ divides the characteristic of $R.$

I want to answer this question: Let $R$ be a ring (with $1$), and let $H$ be the cyclic subgroup of $(R,+)$ generated by $1.$ The characteristic of $R$ is $n$ if $H\cong \mathbb{Z}_{n},$ and is $0$ if ...
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45 views

Conjugates in Field Theory

Let $K/F$ be an algebraic field extension. We say that elements $\alpha$ , $\beta$ $\in$ K are conjugate over F if $\alpha$ and $\beta$ have the same minimal polynomial over F. While reading Field ...
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1answer
36 views

How to understand this statement: A finite field with $n$ elements exists iff $n=p^k$, where $p$ is a prime and $k$ a positive integer

I encountered this statement in a book am reading (but without proof): A finite field with $n$ elements exists iff $n=p^k$, where $p$ is a prime and $k$ is a positive integer. In this case, there is ...
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1answer
46 views

The quotient of the Weyl algebra by its centre is a matrix algebra

Consider the Weyl algebra, call it $W$, generated by $x_1,x_2,...x_n,\partial_1,\partial_2,...\partial_n$ over an algebraically closed field $k$ of characteristic $p>0$ subject to the usual ...
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26 views

Showing that a polynomial splits in a splitting field of another polynomial

Let $k$ be a field with characteristic $p \gt 0$ and let $q(x) = x^p-ax-b \in k[x]$ be a polynomial with $a \neq 0$. Let $K$ be a splitting field of $q(x)$. I have to prove that $x^{p-1}-a$ splits in $...
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1answer
46 views

Morphisms between K3 surfaces

Let $X$ and $Y$ be K3 surfaces over an algebraically closed field of characteristic 0. Is it possible to have a dominant morphism $X\to Y$ which is not an isomorphism? In positive characteristic, the ...
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1answer
36 views

Rank of $R^n$ in characteristic $n$

I'm reading Deligne-Milne's introduction to Tannakian categories, and I noticed a troubling consequence of the definition of rank in a rigid ACU tensor category $(\mathcal{C}, \otimes)$. Specifically, ...
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35 views

Katz's proof of Cartier's (descent) theorem

I am trying to understand the proof of Cartierā€™s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katzā€™s ā€œNilpotent connections and the monodromy theorem: applications of a result of ...
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38 views

In characteristic 2 the splitting field of a cubic has degree 6

I have been working on the following problem, from an old p-set of a Galois theory course I found online: Let $F = \mathbb{F}_2(t)$, the field of rational functions on $\mathbb{F}_2$, and let $f(x) = ...
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1answer
45 views

Is the field characteristic necessary for this proof?

I was looking into this question and I stumbled upon a similar problem, but with slightly different hypothesis: Let $A:M_{n\times n }\left(\mathbb{F}\right)\mapsto M_{n\times n}\left(\mathbb{F}\right)...
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89 views

Bounds on the degree and number of polynomials in the reduced Gröbner basis of an ideal and its radical over a field of positive characteristic.

Disclaimer: Throughout, fix a field $K$ with $\text{char}(K) = p >0$, and we assume that all the computations related to Grƶbner bases are done with a fixed elimination ordering. I am currently ...
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60 views

algebraic de Rham cohomology of blowup of relative line

Let $$T = \mathbb{A}^1_k,\,\,\, Y = \mathbb{P}^1_T,\,\,\, X = \mathscr{B}(Y),$$ the blowup of $Y$ at a point. I am trying to compute the de Rham cohomology $H^1_{dR}(X/T)$, but I could use some help. ...
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148 views

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

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
33 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 $f āˆˆ K[X]$ monic and irreducible with root $\alpha$. Let $F$ be the Frobenius endomorphism. To demonstrate: $\alpha āˆˆ F[K(\alpha)] \Rightarrow f āˆˆ...
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1answer
65 views

$X^{p^k} - a āˆˆ K[X]$ irreducible?

Let $K$ be a field of characteristic char$(K) = p > 0$, and let $a āˆˆ K$ be an element with the following property: $$(\forall \beta āˆˆ K)(\beta^p ā‰  a). $$ Let $k āˆˆ ā„•$ be arbitrarily given. Is it ...
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24 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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58 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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33 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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61 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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34 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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41 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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1answer
44 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
54 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
45 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
54 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
157 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
100 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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47 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
33 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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142 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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1answer
264 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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60 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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1answer
93 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 couldnā€™t any counter example for this claim. Can anyone help me please?
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2answers
128 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
85 views

Transforming Quadrics in Characteristic 2

Iā€™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 $X,Y,Z,T$...

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