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Fields as algebraic objects. For vector and tensor fields, use eg. [dg.differential-geometry]. For physical fields, use eg. [mp.mathematical-physics] or [quantum-field-theory].

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I think I may have a proof of the statement but I am told the statement is false, and can’t find an error in my proof, and am not sure how to construct a counter example. I have attached images of the ...
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Let $A$ be a division ring and $B$ a sub division ring, both noncommutative, and such that $[ A : B] = 2$. Are there examples for which there exists an element $\ell \in A \setminus B$ such that $\ell$...
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Let $f(x)=\sum\limits_{n \ge 0} a_nx^n$ be a power series with $a_n \in \mathbb{C}$ for all $n \ge 0$. Suppose that for all $a \in \bar{\mathbb{Q}}\setminus \{0\}$, we have $f(a) \in \bar{\mathbb{Q}}$....
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The nimbers are the ordinals given an alternative arithmetic structure, turning them into an algebraically complete field of characteristic 2. I've spent the past few months researching the topic. I'...
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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 ...
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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{...
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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 ...
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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\...
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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, ...
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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):...
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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 ...
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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 ...
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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 $\...
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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 ...
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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 ...
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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 ...
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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 ...
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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. ...
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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 ...
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Let $k$ be an infinite field and $K$ a field extension of $k$. Is it possible to intrinsically characterize when $K$ is given by the quotient field of some $k[x_1, \dotsc, x_n]/p$, where $p \subseteq ...
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I'm a teaching tutor in Computer Science, and today after a lecture I spent some time with a student who asked me a very interesting question, in my opinion. We talked about this a bit, then I had to ...
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Consider the extension of fields $K:=\mathbb F_5(T)[y]$ where $y^2=T(T-1)(T-2)(T-3)(T-4)$. It is a Galoisian extension with Galois group $G=\{Id,\sigma\}$ where $\sigma(y)=-y$ and $\sigma(T)=T$. Now ...
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Let $K$ be a field with positive characteristic and $X$ be an indeterminate. Is the extension $K((X))/K(X)$ separable, where $K((X))$ denotes the field of fractions of $K[[X]]$ the ring of formal ...
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Does $(\mathbb{Q},0,1,–,+,×)$ have NIP (Not the Independence Property)? I cannot find an answer let alone a proof on the Internet.
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"Quadratrix" (Wikipedia, 2024-11-14) writes: An accurate construction of the quadratrix would also allow the solution of two other classical problems known to be impossible with compass and ...
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We know that finite fields have prime characteristic and we know a lot about them based in this fact. We can use that knowledge to establish very interesing and deep properties about these fields. In ...
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Reposting from MathStackExchange https://math.stackexchange.com/questions/5028883/determinant-of-an-endomorphism-of-a-drinfeld-module-over-a-finite-field with a slightly more general question. Let $\...
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Let $k$ be a field of characteristic zero and $X$ a smooth proper geometrically integral variety over $k$. Fix a prime number $\ell$. For every integer $n$ and codimension one point $x \in X^{(1)}$ ...
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Intro I have two questions. First, I wonder whether the axioms of a unital composition algebra are "minimal" -- and in particular whether the non-degenerate bilinear form axiom can be ...
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Let $K$ be an imaginary quadratic field. By explicit class field theory, we shall know explicit description of the maximal abelian extension $K^{ab}$ of $K$. In particular, $\mathbb Q(\zeta_n) \...
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The below is from chapter 14.7 of Dummit & Footes "Abstract Algebra" (paraphrased). Theorem: Let $\alpha \in K$ for $K$ a root extension. Then $\alpha$ is contained in a root extension ...
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$\mathbb{Q}(\zeta_{p^\infty})$, also written as $\mathbb{Q}(\mu_{p^\infty})$ or $\mathbb{Q}(p^\infty)$, denotes $\mathbb{Q}$ adjoined with the $p^{n}$th roots of unity for all $n$. It's the union of a ...
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In his paper "A Hasse principle for function fields over PAC fields" (DOI link), Ido Efrat proves the following result: Let $F$ be an extension of a perfect PAC field $K$ of relative ...
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I wonder if there is any classification result on non-archimedean norms on $\mathbb{R}$, with trivial restriction to $\mathbb{Q}$? Any references or examples would be welcomed! Some examples of such ...
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It is well-known that a definable field of finite Morley rank has no proper definable group of automorphisms (a proof can be found for example in the book "Stable groups" of Poizat). My ...
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Let $F$ be a field, let $f \in F[x]$, let $E$ be the splitting field of $f$, and let $e \in E$ be written in terms of the roots of $f$. I'm looking for a simple way to establish if $e \in F$. If $E/F$ ...
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I'm considering the correctness of the following assertion, which is related to linear disjointness (I'm trying to generalize it to subalgebras), What does "linearly disjoint" mean for ...
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It's well known that while there is a natural topological construction of a nearly algebraically closed field of characteristic $0$, algebraically closed fields of positive characteristic seemingly ...
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I ran into this MSE question and would like to ask about its answer and plausible generalizations. The quoted MSE question asks if the following claim is true or false and why: Claim: Let $a,b,c \in \...
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Let $F$ be any field with $\operatorname{Char} F=q$. Let $p$ be a prime such that $p\neq q$. Suppose $F$ has no $p$-th root of unity except $1$. Is it true that the cyclotomic polynomial $X^{p-1}+\...
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Let $\Omega$ be the completion of an algebraic closure of $\mathbb F_q\left(\left(\frac1T\right)\right)$ for the topology induced by the valuation $-\deg$ on $\mathbb F_q(T)$. Let $x,y\in\overline{\...
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In $\mathbb{N}$, we can define an equivalence relation on the Cartesian product $\mathbb{N}^2$ as $(a,b) \sim (c,d)$ if and only if $a + d = b + c$. Then, the quotient set $\mathbb{N}^2 / \sim$ is ...
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This question is basically a special case of this older question of mine, which is still unanswered. Let $\mathcal{D}$ be the field of d.c.e. reals; these turn out to be exactly the reals $\alpha$ for ...
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Let $p$ be an odd prime. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $1\le k\le p-1$ be a divisor of $p-1$. Recently, when I learnt algebraic number theory, I met the following problem. If we let $$U_k=\{...
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This question has been asked in Math.StackExchange (see here) for more than a week and I even put a bounty on it. But still it hasn't been correctly answered (the current answer there was written by ...
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I am currently working on a computer formalization of the algebraic completeness of Conway's nimbers. However, I've found Conway's exposition to be a bit convoluted, and I'm having trouble filling in ...
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Let $L/K$ be an arbitrary finite Galois extension of fields. This induces an injection of Cohen-Witt rings $W_C(K) \to W_C(L)$. Cohen-Witt rings are a generalization of Witt rings - in general they ...
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Let $\ell$ be a division ring, and let $k$ be a sub division ring. I know that a quadratic equation $x^2 + ax + b = 0$, with $a, b \in k$ can have more than two solutions in $\ell$, but what if the ...
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It was showed in "The Stone–Weierstrass Theorem for valuable fields" that the Stone–Weierstrass theorem holds for any topological field whose topology comes from an absolute value or a Krull ...
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Let $f,g \in \mathbb{C}[x,y]$ with total degrees $\deg_{1,1}(f),\deg_{1,1}(g) \geq 1$. Write, $f=a_ny^n+a_{n-1}y^{n-1}+\cdots+a_1y^1+a_0$ and $g=b_my^m+b_{m-1}y^{m-1}+\cdots+b_1y^1+b_0$, for some $n,m ...
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