query: For coprime integers $p,q>0$ and $u,v$ real numbers algebraic over $\mathbb{Q}$ of degrees $p$ & $q$, prove $[\mathbb{Q}(u,v) : \mathbb{Q}] = pq$
positive_passages.docid: 155771
positive_passages.text: Product of degree of two field extensions of prime degree
negative_passages.docid: 113892
negative_passages.text: Show $A \in \cal F$ exists, such that $\mathbb P(A)=1$ and $\mathbb Q(A)=0$.
negative_passages.docid: 138212
negative_passages.text: Please help me prove: $v(a+b)\leq v(a)+v(b)$, and $v(ab)\leq v(a)v(b)$ where $v(x)=\inf{\{\vert x^n \vert}^{1/n}: n\in\mathbb{N}\}$
negative_passages.docid: 73669
negative_passages.text: Totally real Galois extension of given degree
negative_passages.docid: 130944
negative_passages.text: Greatest common divisor of polynomials over $\mathbb{Q}$
negative_passages.docid: 78143
negative_passages.text: Proving $\mathbb{F}_p/\langle f(x)\rangle$ with $f(x)$ irreducible of degree $n$ is a field with $p^n$ elements
negative_passages.docid: 75946
negative_passages.text: Determine the degree of the extension $\mathbb{Q}(\sqrt{3 + 2\sqrt{2}})$.
negative_passages.docid: 118874
negative_passages.text: If $v$ is algebraic over $K(u)$, for some $u\in F$, and $v $ is transcendental over $K$, then $u$ is algebraic over $K(v)$
negative_passages.docid: 151961
negative_passages.text: Irreducibility of $x^{p(p-1)}+x^{p(p-2)}+\cdots+x^p+1$ over $\mathbb Q$
negative_passages.docid: 46730
negative_passages.text: How to show $\mathbb{Q}(\alpha^{4})=\mathbb{Q}(\alpha)$?
negative_passages.docid: 20161
negative_passages.text: Prove that the multiplicative inverse of $a$ modulo $m$ exists if and only if $a$ and $m$ are coprime.
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: 4401
negative_passages.text: Constructive proof that algebraic numbers form a field
negative_passages.docid: 103496
negative_passages.text: Show that $3p^2=q^2$ implies $3|p$ and $3|q$
negative_passages.docid: 134885
negative_passages.text: proving that $\mathbb{Q}(\sqrt{5}, \sqrt{6}) = \mathbb{Q}(\sqrt{5}+ \sqrt{6}) $
negative_passages.docid: 24919
negative_passages.text: $\mathbb{Q}(\zeta_{10}) $ degree of field extension over $\mathbb{Q}$
negative_passages.docid: 103238
negative_passages.text: Conjugates of ${\sqrt 5 + \sqrt 6 }$ over $\mathbb{Q}$ and irreducibility of $x^4 -22x^2+1$ (for minimal polynomial)