Why is $K:=\{x+\sqrt{2}y : x,y\in \mathbb{Q}\}$ a field?
63280
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.
53701
Unique quadratic subfield of $\mathbb{Q}(\zeta_p)$ is $\mathbb{Q}(\sqrt{p})$ if $p \equiv 1$ $(4)$, and $\mathbb{Q}(\sqrt{-p})$ if $p \equiv 3$ $(4)$
31142
Why does $K \leadsto K(X)$ preserve the degree of field extensions?
77862
Question about fields and quotients of polynomial rings
98803
Square roots of integers and cyclotomic fields
134896
Enlightening proof that the algebraic numbers form a field
99516
Show $\mathbb{Q}[\sqrt[3]{2}]$ is a field by rationalizing
47041
What are the conditions for integers $D_1$ and $D_2$ so that $\mathbb{Q}[\sqrt {D_1}] \simeq \mathbb{Q}[\sqrt {D_2}]$ as fields.
54267
Field of fractions of $\mathbb{Q}[x,y]/\langle x^2+y^2-1\rangle$
104300
Help justifying that $\mathbb Q(\sqrt[3]{2})$ is not a splitting field over $\mathbb Q$.
93328
$\mathbb{F}_p[X]/(X^2+X+1)$ is a field iff $p \equiv 2 \bmod 3$
158590
Why is it that if $[L:K]$ is finite, then $[L:K]$ is algebraic?
157496
Determine the Galois group of $\mathbb{Q}(\sqrt{a+b\sqrt{d}})$
31961
Given fields $M/E/F$, why does $[M:F] = [M:E][E:F]$?
33576
Non-principal ideal in $K[x,y]$?
13983
Show that If $R[X]$ is Euclidean domain then $R$ is a field
111291
Galois group of $\mathbb Q(\sqrt{4+\sqrt 7})/\mathbb Q$