query: Simple extension $K( \alpha)$ over $K$ of degree 2, show $ \alpha^2 \in K$
positive_passages.docid: 28803
positive_passages.text: Extension of degree 2 with characteristic not 2
negative_passages.docid: 79679
negative_passages.text: Let $\alpha, \beta \in \mathbb{C}$, such that $\alpha + \beta$, and $\alpha\beta$ are algebraic. Show that $\alpha$ and $\beta$ are algebraic.
negative_passages.docid: 16163
negative_passages.text: Show $\mathbb{Q}(\sqrt{2},\sqrt{3})$ is a simple extension field of $\mathbb{Q}$.
negative_passages.docid: 80538
negative_passages.text: Fields extensions over isomorphic fields of different degrees
negative_passages.docid: 157031
negative_passages.text: Express $x^8-x$ as a product of irreducibles in $\Bbb Z_2[x]$
negative_passages.docid: 102234
negative_passages.text: Show $a^p \equiv a \mod 2p$
negative_passages.docid: 42677
negative_passages.text: Proving $(\alpha + \beta)^{p^n} = \alpha^{p^n} + \beta^{p^n}$ in a finite field
negative_passages.docid: 17276
negative_passages.text: Number of points in the fibre and the degree of field extension
negative_passages.docid: 27577
negative_passages.text: An irreducible polynomial of degree coprime to the degree of an extension is irreducible over this extension
negative_passages.docid: 45302
negative_passages.text: Finding the $\mathbb{Q}$-automorphisms of the splitting field of $x^p-2$ over $\mathbb{Q}$.
negative_passages.docid: 43300
negative_passages.text: The degree of $\sqrt{2} + \sqrt[3]{5}$ over $\mathbb Q$
negative_passages.docid: 13147
negative_passages.text: Proof verification - Minimal polynomial of $\alpha$ has integer coef.s iff its an alg. integer
negative_passages.docid: 622
negative_passages.text: Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$
negative_passages.docid: 106914
negative_passages.text: Existence of Irreducible polynomials over $\mathbb{Z}$ of any given degree
negative_passages.docid: 35216
negative_passages.text: Prove $\mathbb{Q}(\sqrt{2},\sqrt{3},\sqrt{5}):\mathbb{Q}$ is a simple extension
negative_passages.docid: 82824
negative_passages.text: Affine algebraic subset of $\mathbb{A}_k^4$
negative_passages.docid: 143219
negative_passages.text: If a field $F$ is an algebraic extension of a field $K$ then $(F:K)=(F(x):K(x))$