Newbetuts

$K$ is a splitting field $\iff$ any irreducible polynomial with a root in $K$ splits completely over $K$.

Algebraic extensions and sub rings

$F \le E$ extension. every element $\alpha \in E - \overline{F}_E$ over $\overline{F}_E$ transcendental

$\lvert K(\alpha_1,\dots,\alpha_n) : K \rvert$ is a divisor of $n!$

Finite Galois extensions of the form $\frac{\mathbb Z_p[x]}{\langle p(x)\rangle}:\mathbb Z_p$

The exponential extension of $\mathbb{Q}$ is a proper subset of $\mathbb{C}$?

$\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})$ for distinct prime $p,q$

Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$ [duplicate]

"Numbers" bigger than every natural number

How many quadratic extension are there on a field?

So-called Artin-Schreier Extension

Let $x$ be transcendental over $F$. Let $y=f(x)/g(x)$ be a rational function. Prove $[F(x):F(y)]=\max(\deg f,\deg g)$

Does there exist a Hamel basis $\mathcal B$ for $\mathbb R$ over $\mathbb Q$ such that $a,b \in \mathcal B \implies \dfrac ab \in \mathcal B$?

Why are $i$ and $-i$ "more indistinguishable" than $\sqrt{2}$ and $-\sqrt{2}$?

How to show that $\mathbb{Q}(\sqrt{p},\sqrt{q}) \subseteq \mathbb{Q}(\sqrt{p}+\sqrt{q})$

$K(u,v)$ is a simple extension of fields if $u$ is separable

$\mathbf{Q}[\sqrt 5+\sqrt[3] 2]=\mathbf{Q}[\sqrt 5,\sqrt[3] 2]$?

New posts in extension-field

If a field $F$ is an algebraic extension of a field $K$ then $(F:K)=(F(x):K(x))$

Find intermediate fields of $\mathbb{Q}(\sqrt[3]{2}, \sqrt{3},i) \, | \, \mathbb{Q}(i)$

Understanding that $\mathbb{R}(X^2 + Y^2, XY)(x) \supset \mathbb{R}(Y)$?

$K$ is a splitting field $\iff$ any irreducible polynomial with a root in $K$ splits completely over $K$.

Algebraic extensions and sub rings

$F \le E$ extension. every element $\alpha \in E - \overline{F}_E$ over $\overline{F}_E$ transcendental

$\lvert K(\alpha_1,\dots,\alpha_n) : K \rvert$ is a divisor of $n!$

Finite Galois extensions of the form $\frac{\mathbb Z_p[x]}{\langle p(x)\rangle}:\mathbb Z_p$

The exponential extension of $\mathbb{Q}$ is a proper subset of $\mathbb{C}$?

$\mathbb{Q}(\sqrt{p},\sqrt[3]{q})=\mathbb{Q}(\sqrt{p}\sqrt[3]{q})$ for distinct prime $p,q$

Show that $\mathbb Q(\sqrt p) \not\simeq\mathbb Q(\sqrt q)$ [duplicate]

"Numbers" bigger than every natural number

How many quadratic extension are there on a field?

So-called Artin-Schreier Extension

Let $x$ be transcendental over $F$. Let $y=f(x)/g(x)$ be a rational function. Prove $[F(x):F(y)]=\max(\deg f,\deg g)$

Does there exist a Hamel basis $\mathcal B$ for $\mathbb R$ over $\mathbb Q$ such that $a,b \in \mathcal B \implies \dfrac ab \in \mathcal B$?

Why are $i$ and $-i$ "more indistinguishable" than $\sqrt{2}$ and $-\sqrt{2}$?

How to show that $\mathbb{Q}(\sqrt{p},\sqrt{q}) \subseteq \mathbb{Q}(\sqrt{p}+\sqrt{q})$

$K(u,v)$ is a simple extension of fields if $u$ is separable

$\mathbf{Q}[\sqrt 5+\sqrt[3] 2]=\mathbf{Q}[\sqrt 5,\sqrt[3] 2]$?