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 $I(x)=x$?

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 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}| […]

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 […]

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 […]

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?

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 […]

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 […]

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