Articles of positive characteristic

the positive square root of $I$?

Find operators $T \colon \mathbb{R}^2 \to \mathbb{R}^2$ such that $T^2=I$. Which one is the positive square root of $I$? Is there the operator $T$ such that $T(T(x))=I$ where $I(x)=x$?

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 $n$, are there subsets $J \subset \{1, \ldots n\} \times \{1, \ldots, n\}$ […]

Let $R$ be a commutative unital ring. Is it true that the group of units of $R$ is not isomorphic with the additive group of $R$?

Let $R$ be a commutative ring with unity, and let $R^{\times}$ be the group of units of $R$. Then is it true that $(R,+)$ and $(R^{\times},\ \cdot)$ are not isomorphic as groups ? I know that the statement is true in general for fields. And it is trivially true for any finite ring (as $|R^{\times}| […]

Show that $f(x) = x^p -x -1 \in \Bbb{F}_p$ 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 $\alpha \in \Bbb{F}_p$, $\alpha^{p-1} = 1 \implies \alpha^p=\alpha \implies \alpha^p-\alpha=0$. Of course $0^p=0$, so this is true for all elements of $F$ and not just the […]

Multiplication by $m$ isogenies of elliptic curves in characteristic $p$

I’ve been attempting to prove some comments I’ve read on MO by myself for my undergrad thesis regarding étale morphisms of elliptic curves. My definition of an étale morphism is taken from Milne’s notes (page 19) which roughly says that for a field $K$ with algebraic closure $\bar{K}$, a morphism $\phi: X\rightarrow Y$ of varieties […]

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. Any ideas?

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 $c\in F$. $\bf{Proof:}$ First of all we note that $f'(x)=-1$, hence $\gcd(f,f’)=1$, and we […]

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 a contradiction they are, then any isomorphism sends solutions […]