Intereting Posts

Asymptotic expansion of the integral $\int_0^1 e^{x^n} dx$ for $n \to \infty$
Characterization of Harmonic Functions on the Punctured Disk
Does there exist coprime numbers $a$ and $b$ such that $a^n+b$ is composite for every $n$?
Indefinite Integral $\int\sqrt{\tan(x)}dx$
Is this:$\sum_{n=1}^{\infty}{(-1)}^{\frac{n(n-1)}{2}}\frac{1}{n}$ a convergent series?
Can the determinant of an integer matrix with $k$ given rows equal the gcd of the $k\times k$ minors of those rows?
a Circle perimeter as expression of $\pi$ Conflict?
Simple Proof of the Euler Identity $\exp{i\theta}=\cos{\theta}+i\sin{\theta}$
Is there a $p$-adic version of Liouville theorem?
Closed-form of an integral involving a Jacobi theta function, $ \int_0^{\infty} \frac{\theta_4^{n}\left(e^{-\pi x}\right)}{1+x^2} dx $
Continuous injective map $f:\mathbb{R}^3 \to \mathbb{R}$?
How to prove that $\frac1{n\cdot 2^n}\sum\limits_{k=0}^{n}k^m\binom{n}{k}\to\frac{1}{2^m}$ when $n\to\infty$
Countable axiom of choice: why you can't prove it from just ZF
Yoneda-Lemma as generalization of Cayley`s theorem?
How to determine the arc length of ellipse?

What are some examples of algebraically closed fields? Wikipedia lists exactly two: $\mathbb{C}$ and the (complex) algebraic numbers.

EDIT: scrolling to the bottom of the Wikipedia article, they mention that every field has an essentially unique “algebraic closure”, which is algebraically closed, and that proving this fact in full generality requires the Axiom of Choice.

But that leaves open some questions: are there any algebraically closed fields that are used often enough that they have names? When do you need AC to show $\mathbb{F}$ has an algebraic closures, and when don’t you need AC?

- Finite Generated Abelian Torsion Free Group is a Free Abelian Group
- $\mathbb Z^n/\langle (a,…,a) \rangle \cong \mathbb Z^{n-1} \oplus \mathbb Z/\langle a \rangle$
- $M' \to M \to M'' \to 0$ exact $\implies 0\to \text{Hom}(M'',N) \to \text{Hom}(M,N) \to \text{Hom}(M',N)$ is exact.
- Non abelian group with normal subgroup
- Galois Group of $\sqrt{1+\sqrt{2}}$
- If $=n$, then $g^{n!}\in H$ for all $g\in G$.

I could not find a similar question here, and Google was not helpful.

- If $K^{\mathrm{Gal}}/F$ is obtained by adjoining $n$th roots, must $K/F$ be as well?
- Linearly dependent polynomials
- Subring of a finitely generated Noetherian ring need not be Noetherian?
- Representing Elementary Functions in a CAS
- Prove that $G$ is abelian iff $\varphi(g) = g^2$ is a homomorphism
- Show that Pn is an (n+1)-dimensional subspace
- Question regarding isomorphisms in low rank Lie algebras
- Proof that an affine scheme is quasi compact
- What is the interpretation of $a \equiv b$ mod $H$ in group theory?
- Finite dimensional algebra with a nil basis is nilpotent

In general, an algebraic closure of a field $K$ is denoted by $\overline{K}$. Typical examples arising in number theory are $K=\mathbb{Q}$, $K=\mathbb{F}_p(t)$, $K=\mathbb{Q}_p$. Usually one needs the axiom of choice in order to prove the existence of algebraic closures. There are (at least) two exceptions: For $K=\mathbb{R}$ we have $\overline{K}=\mathbb{C}$. Then, for every subfield $K \subseteq \mathbb{R}$, we may realize $\overline{K}$ as the subfield of $\mathbb{C}$ which consists of complex numbers which are algebraic over $K$. It also exists without AC. We may also replace $\mathbb{R}$ by a real closed field, one only has to adjoin $\sqrt{-1}$. For $K=\mathbb{F}_q$, a finite field, we have for every $n$ an extension $\mathbb{F}_{q^n}$ of degree $n$ and every divisibility relation $n|m$ induces a canonical $\mathbb{F}_q$-homomorphism $\mathbb{F}_{q^n} \to \mathbb{F}_{q^m}$. It follows that we may consider the *colimit* $\mathbb{F}_{q^{\infty}} := \varinjlim_{n} \mathbb{F}_{q^n}$ (often this is written as a union, which is not quite correct). This turns out to be an algebraic closure of $\mathbb{F}_q$.

Let me also share a quite nice construction of an algebraic closure: Consider the (infinite) tensor product $A$ of all the $K$-algebras $K[x]/(f)$, where $f \in K[x] \setminus \{0\}$. By linear algebra it is non-zero, hence has a maximal ideal $\mathfrak{m}$ (Zorn’s Lemma!). Then $K’ := A/\mathfrak{m}$ is a field extension of $K$, and by construction every $f \in K[x] \setminus \{0\}$ has a root in $K’$. It is a nontrivial result that this already *is* the algebraic closure; but even if we don’t use this, we can just repeat this process $K \hookrightarrow K’ \hookrightarrow K” \hookrightarrow K”’ \hookrightarrow \dotsc$ and observe that the colimit $\overline{K}$ is an algebraic closure of $K$. A similar reasoning can be obtained to show that every two algebraic closures are isomorphic (but not in a canonical way).

- Additive rotation matrices
- Example of a Problem Made Easier with Skew Coordinates
- Why are Vandermonde matrices invertible?
- Show the Volterra Operator is compact using only the definition of compact
- Integrate and measure problem.
- Infinitesimals – what's the intuition?
- Possible road-maps for proving $\lim_{x\to 0}\frac{\sin x}{x}=1$ in a non-circular way
- Derivation of the general forms of partial fractions
- Proving the Law of the Unconscious Statistician
- How to make a smart guess for this ODE
- ODE introduction textbook
- Probability of tossing a fair coin with at least $k$ consecutive heads
- How to use the Mean Value Theorem to prove the following statement:
- Any Set with Associativity, Left Identity, Left Inverse is a Group
- Why the morphisms of vector spaces, over different fields is not interesting?