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
0 votes
0 answers
15 views

Separability of the Frobenius morphism

We are in an algebraically closed field $k$ with positive characteristic $p>0$. Let $X\subseteq \mathbb{P}^2$ be an irreducible algebraic curve and $\phi:X\to\phi(X),\ (x_0,x_1,x_2)\mapsto (x_0^p,...
user avatar
1 vote
1 answer
44 views

Irreducible polynomial in field of positive characteristic

Let $F$ be a field of characteristic $p > 0$ and $C$ be an element of $F$ that is not a $p$-power. For a positive integer $s$, show that $x^{p^s} − C$ is irreducible, and its splitting field is of ...
user avatar
  • 463
5 votes
0 answers
114 views

Over any field, if some polynomials are algebraically dependent, are their derivatives linearly dependent?

Denote by $P_n$ the space of all polynomials in $n$ variables, with coefficients in a field $\mathbb F$. A collection of polynomials $(f_1,\cdots,f_m)=\vec f\in P_n\!^m$ is called algebraically ...
user avatar
  • 4,150
0 votes
1 answer
71 views

Why supposing that $\operatorname{char}(F) \neq 2$?

I have a question regarding to part$(b)$ in the problem below: " Let $V$ be a finite-dimensional vector space over the field $F,$ and let $B$ be a bilinear form on V. $(b)$ Suppose $\operatorname{...
user avatar
  • 1,163
3 votes
0 answers
68 views

Monogenic function fields

Recall that a number field $K$ is said to be monogenic if its ring of integers is of the form $\mathbb{Z}[\alpha]$, or equivalently, if it has a power integral basis. There are many references one can ...
user avatar
  • 73
4 votes
2 answers
346 views

Example of ring can’t be defined over finite field.

Let $K$ be an infinite field of characteristic $p>0$, and $A$ be a $K$-algebra. We say that $A$ can be defined over a finite field if there is finite subfield $F\subset K$ and an $F$-algebra $A_0$ ...
user avatar
  • 936
0 votes
0 answers
42 views

Regularity vs smoothness in positive characteristic

It is well known that a scheme over a perfect field is smooth at $x$ if and only if it is regular at $x$, and that these two properties are not equivalent over non-perfect fields. What is an example ...
user avatar
  • 785
6 votes
1 answer
88 views

Algebraic Peter-Weyl in positive characteristic

To my understanding there is an algebraic version of Peter-Weyl that holds in characteristic $0$ that says for any reductive group $G$ one has that: $$k[G]=\bigoplus V\otimes V^*$$ as a $G\times G$-...
user avatar
  • 195
1 vote
1 answer
55 views

Proof verification of the theorem that integral domains have either $0$ or prime characteristic

I am doing Contemporary Abstract Algebra by Gallian and am stuck at the proof of the fact that integral domains have either $0$ or prime characteristic. The proof in the book goes as follows: Let $R$ ...
user avatar
2 votes
0 answers
88 views

General position theorem for curves: do we use characteristic zero in this proof?

I'm reading about the general position theorem in Arabello, Cornalba, Griffiths, and Harris' Geometry of Algebraic Curves volume 1. Everything is over $\Bbb C$, and for these authors a curve is a ...
user avatar
  • 2,159
2 votes
0 answers
31 views

Prove an element not belonging to the tight closure of an ideal

I'm working on a ring $R=\mathbb{F}_7[x,y,z]/(x^2+y^3+z^5)$ and want to prove $x\notin(y,z)^*$, the tight closure. First I want to find a test element, which can be obtained from the Jacobian ideal. ...
user avatar
  • 2,423
1 vote
0 answers
40 views

Surjective proper cover by ordinary varieties

Let $X$ be smooth proper variety over a finite field $k$ of positive characteristic $p$. Assume that $X$ is not ordinary, then my question is if there exists a smooth, projective ordinary $Y$ with a ...
user avatar
3 votes
1 answer
55 views

Castelnuovo's contractibility criterion in $\text{char}(k)>0$?

Let $S$ be a smooth, projective surface over an algebraically closed field $k$ of any characteristic. I'm trying to prove/disprove the following: There cannot be a sequence of curves $\{E_n\}_{n\in\...
user avatar
  • 8,733
0 votes
0 answers
28 views

Characteristic of a domain

I was looking at a quiz that is not a homework, an exercise or something I should submit. The quiz says to choose the statement that is always true : Let $R$ be a domain (ring with no nonzero zero-...
user avatar
  • 996
2 votes
1 answer
91 views

why do matrices in orthogonal group have determinant 1 in characteristic 2?

Let $q$ be quadratic form on $V=K^n$ where $K$ is a field in characteristic $2$. Let $b(x,y)=q(x+y)-q(x)-q(y)$ be the bilinear form associated to $q$ and set $V^\perp=\{x\in V\mid b(x,y)=0 \text{ for ...
user avatar
  • 21
2 votes
1 answer
108 views

Why is the Levi-decomposition not available in positive characteristic?

I am learning about Lie-Algebras and I came across a proof of the Levi-decomposition. Can somebody tell why the Lie-Algebra has to be over a field of characteristic $0$? Why does it fail for positive ...
user avatar
  • 47
2 votes
0 answers
25 views

Find invariant ring in modular case

I'm learning Invariant Theory of Finite Groups and saw this problem: "What is $S^G$ where $S := \mathbb{F}_p[x, y]$, and $G$ is the subgroup of $GL_2(\mathbb{F}_p)$ generated by $\begin{pmatrix} ...
user avatar
  • 2,423
1 vote
1 answer
36 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(\...
user avatar
1 vote
0 answers
55 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 ...
user avatar
  • 1,474
3 votes
1 answer
79 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 ...
user avatar
  • 1,497
5 votes
0 answers
50 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 ...
user avatar
3 votes
0 answers
119 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\...
user avatar
  • 547
2 votes
0 answers
142 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+...
user avatar
  • 664
1 vote
0 answers
81 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 ...
user avatar
  • 7,809
2 votes
0 answers
41 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 ...
user avatar
  • 1,012
3 votes
2 answers
166 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 $x^2-b$. Provide ...
user avatar
1 vote
1 answer
73 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 ...
user avatar
2 votes
1 answer
136 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, ...
user avatar
  • 1,356
0 votes
0 answers
29 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{...
user avatar
0 votes
1 answer
48 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 ...
user avatar
1 vote
0 answers
64 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[\...
user avatar
  • 11
0 votes
1 answer
116 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 ...
user avatar
1 vote
0 answers
60 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 ...
user avatar
1 vote
1 answer
75 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 ...
user avatar
  • 705
0 votes
1 answer
77 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 ...
user avatar
0 votes
0 answers
28 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 $...
user avatar
1 vote
1 answer
69 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 ...
user avatar
0 votes
1 answer
63 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, ...
user avatar
  • 785
2 votes
0 answers
52 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 ...
user avatar
2 votes
0 answers
89 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) = ...
user avatar
2 votes
1 answer
57 views

Is the field characteristic necessary for this diagonalization question?

I was looking into this question and I stumbled upon a similar problem, but with a slightly different hypothesis: Let $A:M_{n\times n }\left(\mathbb{F}\right)\mapsto M_{n\times n}\left(\mathbb{F}\...
user avatar
  • 5,030
4 votes
0 answers
128 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 ...
user avatar
  • 1,856
3 votes
0 answers
82 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. ...
user avatar
1 vote
2 answers
260 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?
user avatar
  • 688
1 vote
1 answer
50 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 ∈...
user avatar
1 vote
1 answer
161 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 ...
user avatar
0 votes
0 answers
29 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 ...
user avatar
  • 1,144
0 votes
0 answers
86 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 ...
user avatar
0 votes
0 answers
41 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 ...
user avatar
  • 118
2 votes
0 answers
85 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 $X$....
user avatar