Newbetuts

Show that the set $\mathbb{Q}[\sqrt{2}] = \{a + b \sqrt{2} \mid a, b \in \mathbb{Q}\}$ is a field with the usual multiplication and addition.

Simple property of a valuation on a field

Embedding Fields in Matrix Rings

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Finiteness of the Algebraic Closure

Is number rational?

Extended Euclidean Algorithm in $GF(2^8)$?

How to prove that a complex number is not a root of unity?

Is $\mathbf{C}$ the algebraic closure of any field other than $\mathbf{R}$?

Why must be the additive and multiplicative identities in a field be different?

Find the splitting field of $x^4-4x^2+1$ over $\mathbb{Q}$

Show that if the field of $p^a$ elements is a subfield of the field of $p^b$ elements if and only if $a\vert b$.

$f(x) $ be the minimal polynomial of $a$ (algebraic element) over $\mathbb Q$ , let $b=f'(a) \in \mathbb Q(a)$ , then is $\mathbb Q(a)=\mathbb Q(b)$?

Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$ [duplicate]

Irreducibility of a polynomial if it has no root (Capelli) [duplicate]

New posts in field-theory

Prove that both $x+y$ and $xy$ are rational, under some conditions

Why algebraic closures?

What fields between the rationals and the reals allow to define the usual 2D distance?

On the meaning of being algebraically closed

What is the coproduct of fields, when it exists?

Show that the set $\mathbb{Q}[\sqrt{2}] = \{a + b \sqrt{2} \mid a, b \in \mathbb{Q}\}$ is a field with the usual multiplication and addition.

Simple property of a valuation on a field

Embedding Fields in Matrix Rings

Does there exist two non-constant polynomials $f(x),g(x)\in\mathbb Z[x]$ such that for all integers $m,n$, gcd$(f(m),g(n))=1$?

Finiteness of the Algebraic Closure

Is number rational?

Extended Euclidean Algorithm in $GF(2^8)$?

How to prove that a complex number is not a root of unity?

Is $\mathbf{C}$ the algebraic closure of any field other than $\mathbf{R}$?

Why must be the additive and multiplicative identities in a field be different?

Find the splitting field of $x^4-4x^2+1$ over $\mathbb{Q}$

Show that if the field of $p^a$ elements is a subfield of the field of $p^b$ elements if and only if $a\vert b$.

$f(x) $ be the minimal polynomial of $a$ (algebraic element) over $\mathbb Q$ , let $b=f'(a) \in \mathbb Q(a)$ , then is $\mathbb Q(a)=\mathbb Q(b)$?

Prove $\sqrt[3]{3} \notin \mathbb{Q}(\sqrt[3]{2})$ [duplicate]

Irreducibility of a polynomial if it has no root (Capelli) [duplicate]