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.

Filter by
Sorted by
Tagged with
1
vote
1answer
19 views

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(\...
1
vote
0answers
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 ...
3
votes
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 ...
5
votes
0answers
35 views

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 ...
3
votes
0answers
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\...
2
votes
0answers
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+...
1
vote
0answers
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 ...
2
votes
0answers
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 ...
0
votes
0answers
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 $...
3
votes
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 $(x^2-b)$. ...
1
vote
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 ...
1
vote
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, ...
0
votes
0answers
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{...
0
votes
1answer
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 ...
1
vote
0answers
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[\...
0
votes
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 ...
1
vote
0answers
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 ...
1
vote
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 ...
0
votes
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 ...
0
votes
0answers
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 $...
1
vote
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 ...
0
votes
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, ...
2
votes
0answers
35 views

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

I am trying to understand the proof of Cartier’s theorem on pages 370-371 (pages 17-18 of the PDF file) of Katz’s “Nilpotent connections and the monodromy theorem: applications of a result of ...
2
votes
0answers
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) = ...
2
votes
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)...
4
votes
0answers
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 Gröbner bases are done with a fixed elimination ordering. I am currently ...
3
votes
0answers
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. ...
1
vote
2answers
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?
1
vote
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 $f ∈ K[X]$ monic and irreducible with root $\alpha$. Let $F$ be the Frobenius endomorphism. To demonstrate: $\alpha ∈ F[K(\alpha)] \Rightarrow f ∈...
1
vote
1answer
65 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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
0
votes
0answers
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 ...
2
votes
0answers
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 $X$....
2
votes
0answers
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 ...
1
vote
0answers
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 ...
1
vote
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 $b$, ...
2
votes
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 $f(x)$ ...
1
vote
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 ...
3
votes
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 ...
3
votes
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 ...
4
votes
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 ...
1
vote
0answers
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 ...
0
votes
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 ...
4
votes
0answers
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 $k$-...
4
votes
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 ...
1
vote
0answers
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)^{...
-2
votes
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 couldn’t any counter example for this claim. Can anyone help me please?
1
vote
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 ...
1
vote
1answer
85 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 $X,Y,Z,T$...