The following fact is well-known, and not hard to prove, but I do not know an explicit reference.Let $R$ be the subring of complex numbers generated by all roots of unity. Then $R$ is free as an ...

Suppose that $K/k$ is a transcendental field extension with both fields algebraically closed (I'm most interested in the case when $\text{char}(k) > 0$). Consider the automorphism group $G = \text{...

When we define a group homomorphism $\theta \colon G \to H$, we do not have to specify that $\theta(e_G) = e_H$. On the other hand, most literature defines a ring homomorphism $h \colon R \to S$ with ...

Everything is in positive characteristic. I am in the following setting: L/K is a finite field extension and I have a morphism $\varphi:K \rightarrow K$ such that $K/\varphi(K)$ is finite and $L\...

Each nontrivial involutory automorphism of $\mathbb{C}$ gives rise to an elementwise fixed subfield $\mathbb{K}$ which is real-closed, and such that $[\mathbb{C} : \mathbb{K}] = 2$, and as we know, ...

Definition (Chowla subspace).Let $K \subseteq L$ be a field extension and let $A$ be a $K$-subspace of $L$.We say that $A$ is a Chowla subspace if for every $a \in A \setminus \{0\}$ one has$$[K(a):...

It is a famous open problem in field arithmetic whether $\mathbb{Q}^{\mathrm{solv}}$, the solvable closure of $\mathbb{Q}$, is pseudo algebraically closed (PAC). That is, whether every absolutely ...

An amusing observation is that there are actually a fair number of familiar rings that satisfy the axioms of Peano arithmetic (in the language $\{+,\cdot,0,1\}$) except for the assertion that $0$ is ...

Let $L/K$ be a field extension, and let $M$ be a (finite, say) Galois module over $K$. Given an automorphism $\sigma : L \to L$, we can lift to an automorphism $\bar \sigma$ of the algebraic closure $\...

Assume $K$ a field not algebraically closed (in reality the isomorphisms follows quite easily if $K$ is closed for quadratic extensions). Define $\mathfrak{sl}_2(K)$ the Lie algebra over $K$ given by ...

It is known that there are infinitely many Polish topologies that make $(\mathbb{R},+)$ into a topological group. However what about $(\mathbb{R}, +, \times)$? Is the standard one the only Polish ...

Let $D$ be a division ring of dimension $n$ over the center $Z(D)$, and let $m$ be the dimension of a maximal subfield $F$ over $Z(D)$.If $n$ is finite, then it is a square $n = a^2$ (with $a$ a ...

Let $\ell$ be a division ring of left dimension $2$ (as a vector space) over the sub division ring $k$.Suppose that all quadratic equations $x^2 + ax + b = 0$ with $a, b \in k$, either have no root ...

A very important problem in the intersection of (birational) algebraic geometry (function fields), algebra (field theory) and logic is the Elementary Equivalence vs Isomorphism Problem of Fields. ...

Given a field $F$ and an automorphism $\phi:F\to F$, according to the definition in Kedlaya's book, a strongly difference closed field means that any dualizable (i.e., admits a dual) difference module ...