Can every quadratic extension of $\mathbb{Q}$ be extended to a cyclic extension of degree 4?
24168
A characterisation of quadratic extensions contained in cyclic extensions of degree 4
41815
Galois extension of intersection of fields
16163
Show $\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a simple extension field of $\mathbb{Q}$.
46730
How to show $\mathbb{Q}(\alpha^{4})=\mathbb{Q}(\alpha)$?
111347
Finding an example of field extensions satisfying certain properties
83595
Proving that $\mathbb{A}=\{\alpha \in \mathbb{C}: \alpha \text{ is algebraic over } \mathbb{Q} \}$ is not a finite extension
622
Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$
111444
Is any finite-dimensional extension of a field, say $F$, algebraic and finitely generated?
132386
Show that $(\mathbb{Q}^*,\cdot)$ and $(\mathbb{R}^*,\cdot)$ aren't cyclic
64878
Is this polynomial irreducible in $\mathbb{Q}$?
99835
Polynomial in $\mathbb{Q}[x]$ with root $\sqrt[3]{2}+\sqrt[3]{3}$
132725
Is $\mathbb{Z}/\langle n\rangle$ cyclic and of order $n$?
104634
Galois extension is (not) transitive
67001
Factorizing polynomials over fields other than $\mathbb{C}$
103654
A finite field extension that is not simple
141184
Proof that a degree 4 polynomial has at least two roots
19525
Showing field extension $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})/\mathbb{Q}$ degree 8