Difficulty in proving that Q[z] is a field if z is an algebraic complex number
58424
If $A$ an integral domain contains a field $K$ and $A$ over $K$ is a finite-dimensional vector space, then $A$ is a field.
46730
How to show $\mathbb{Q}(\alpha^{4})=\mathbb{Q}(\alpha)$?
9129
Show that $\mathbb{Q}(\zeta_n)$ is Galois over $\mathbb{Q}$ and $Gal(\mathbb{Q}(\zeta_n)/\mathbb{Q}))\cong \mathbb{Z_n}^*$
137858
Showing that $\mathbb{Z}[i]/I$ is a finite field whenever $I$ is a prime ideal, and also finding its cardinality?
19525
Showing field extension $\mathbb{Q}(\sqrt{2}, \sqrt{3}, \sqrt{5})/\mathbb{Q}$ degree 8
109647
Ext$_R^n(Q,A)=0=$Tor$_n^R(Q,A)$ where $Q$ is the field of fractions of a domain $R$
53432
What is the algebraic closure of $\mathbb F_q$?
4401
Constructive proof that algebraic numbers form a field
5032
Compute $e^{e^z}$, where z is a complex number
33586
Prove the field of fractions of $F[[x]]$ is the ring $F((x))$ of formal Laurent series.
54267
Field of fractions of $\mathbb{Q}[x,y]/\langle x^2+y^2-1\rangle$
24224
How to prove the ring of Laurent polynomials over a field is a principal ideal domain?
51021
Find all the intermediate fields of the splitting field of $x^4 - 2$ over $\mathbb{Q}$
31451
Proving a subring of $\mathbb{Q}$ containing $\mathbb{Z}$ is a PID
73713
Prove that $\mathbb{Q}(\sqrt{(2+\sqrt{2})(3+\sqrt{3})})$ is Galois over $\mathbb{Q}$ is Galois and determine the Galois group
64130
Prove that $\mathbb{Q}^{\times}$ not isomorphic to $\mathbb{Z}^{n}$
58009
How to show that $\mathbb Q(\sqrt 2)$ is not field isomorphic to $\mathbb Q(\sqrt 3).$