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.

In the category of rings with unity and unity-preserving ring homomorphisms, the integers are an initial object. This means for each ring $R$, there is a unique map $\mathbb{Z}\to R$. If the map is injective, we say $R$ has characteristic $0$. If this map is not injective, its kernel must have the form $n\mathbb{Z}$ for some natural number $n$. In this case, we say that the characteristic of $R$ is $n$, and we write $\operatorname{char}R=n$. Typical examples of such rings are the finite fields along with their extensions. This tag is for questions involving rings such that $\operatorname{char}R=n>0$.

Topics such as representation theory and algebraic geometry have very different tastes depending on the characteristic of the base field $k$. For example, certain foundational theorems in characteristic $0$ are either open questions in positive characteristic, or known to be false. See resolution of singularities and Maschke's theorem.

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How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$?

Let $p$ be a prime. How do I prove that $x^p-x+a$ is irreducible in a field with $p$ elements when $a\neq 0$? Right now I'm able to prove that it has no roots and that it is separable, but I have not a clue as to how to prove it is irreducible.…
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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 snappiest proofs I know is that, if we suppose for…
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Does the category of algebraically closed fields of characteristic $p$ change when $p$ changes?

EDIT I've now posted this question on mathoverflow. It probably makes sense to post answers over there, unless someone prefers posting here. Let $\mathrm{ACF}_p$ denote the category of algebraically closed fields of characteristic $p$, with all…
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How can a field have a finite characteristic $p$, given that a field has no zero divisors?

The characteristic of a field is defined to be the smallest positive integer $p$ such that $$p \cdot 1 = 0.$$ But I have learned that field has no zero divisors. How is this possible?
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Problem in Jacobson's Basic Algebra (Vol. I)

It is section $4.4$, exercise number $5$. It says the following: Let $F$ be a field of characteristic $p$. Show that $f(x)=x^p-x-a$ has no multiple roots and that $f(x)$ is irreducible in $F[x]$ if and only if $a\neq c^p-c$ for any $c\in F$.…
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Characteristic of a finite ring with $34$ units

Let $R$ be a finite ring such that the group of units of $R$, $U(R)$, has $34$ elements. I would like to find the characteristic of $R$. Let $k:= \mathrm{Char}(R)$. If $\varphi$ denotes the Euler totient, then $\varphi(k)$ divides $34$, hence…
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$x^p-x+a$ irreducible for nonzero $a\in K$ a field of characteristic $p$ prime

Is it true that $f(x)=x^p-x+a\in K[x]$ is irreducible for nonzero $a\in K$ a field of characteristic $p$ prime? I've seen variants of this question around, but they don't seem to answer the question as worded. (It's possible I have not searched…
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Show that $f(x) = x^p -x -1 \in \Bbb{F}_p[x]$ is irreducible over $\Bbb{F}_p$ for every $p$.

Let $p$ be a prime. a) Show that $f$ has no roots in $\Bbb{F}_p$. Let $F^*$ be the multiplicative group of $\Bbb{F}_p$. Then, by lagrange's thoerem for all nonzero $\alpha \in \Bbb{F}_p$, $\alpha^{p-1} = 1 \implies \alpha^p=\alpha \implies…
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Can a ring of positive characteristic have infinite number of elements?

For curiosity: can a ring of positive characteristic ever have infinite number of distinct elements? (For example, in $\mathbb{Z}/7\mathbb{Z}$, there are really only seven elements.) We know that any field/ring of characterisitc zero must have…
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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. Determine all $P(X)\in K[X]$ such that…
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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 $\hat{\phi}$ satisfies several nice properties. Two of…
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Why is $W_n(k)$ the unique flat lifting of a perfect field $k$ over $\mathbf{Z}/p^n$?

Let $k$ be a perfect field of characteristic $p>0$ and denote by $W_n(k)$ the ring of Witt vectors over $k$ of length $n$. In their article on the decomposition of the de Rham complex, Deligne and Illusie claim that $W_n(k)$ is the unique flat…
Nuno
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What is an algebraically closed field of characteristic $p$?

I suspect that this is a very simple question, but I need to ask. My question is How do the fields of characteristic $p$ look like? If $K$ is a finite field of order $p^n$, then $K$ has characteristic $p$ ($p$ prime). We can take the algebraic…
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Question about matrices whose row and column sums are zero

I am interested in $n \times n$ matrices over some field $K$ all whose rows and all whose columns sum to zero. First question: do these matrices have a name? Pending an answer I will call these "null-matrices". Second (main) question: Given $n$,…
Vincent
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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 order $p^m$, one uses Artin-Schreier-Witt…
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