query: Is it true that $k[\alpha,\beta]=k(\alpha,\beta)$, $\alpha$ and $\beta$ are algebraic over $k$.
positive_passages.docid: 145507
positive_passages.text: Is $K[a,b]=K(a,b)$ for algebraic $a, b$?
negative_passages.docid: 68790
negative_passages.text: Do we have $K(\beta)=K(\beta^2)$ for field extension of odd degree?
negative_passages.docid: 156679
negative_passages.text: $K[X,Y]/(XY-1)$ not isomorphic to $K[T]$
negative_passages.docid: 33576
negative_passages.text: Non-principal ideal in $K[x,y]$?
negative_passages.docid: 40162
negative_passages.text: Prove that the set $\mathbb{Q}[\alpha] = \{a+b\alpha+c\alpha^2 \mid a, b, c \in\mathbb{Q}\}$ is closed under addition and multiplication.
negative_passages.docid: 142168
negative_passages.text: can we a prove ⊢ (α → α) → (α → α)
negative_passages.docid: 45760
negative_passages.text: if G is generated by {a,b} and ab=ba, then prove G is Abelian
negative_passages.docid: 148310
negative_passages.text: Showing rational numbers are algebraic
negative_passages.docid: 157488
negative_passages.text: Proving that $k[x,y,z]/(xz - (y^2 + 1))$ is a normal ring.
negative_passages.docid: 43219
negative_passages.text: If $[K(\alpha):K]=p\neq q=[K(\beta):K]$ then $[K(\alpha+\beta):K]=pq$
negative_passages.docid: 34714
negative_passages.text: What is the relationship between $\ker(A), \ker(A^2), \ldots, \ker(A^n)$?
negative_passages.docid: 75639
negative_passages.text: Showing that $\sqrt[3]{2}\notin\Bbb Q(\alpha_1,...,\alpha_k)$ where $\alpha_i^2\in\Bbb Q\ \forall i$
negative_passages.docid: 42652
negative_passages.text: Must $k$-subalgebra of $k[x]$ be finitely generated?
negative_passages.docid: 105170
negative_passages.text: About product of $k$ commutators
negative_passages.docid: 137983
negative_passages.text: Normalisation of $k[x,y]/(y^2-x^2(x-1))$
negative_passages.docid: 8087
negative_passages.text: Algebraic integers of $\mathbb{Q}(\sqrt{m})$ for $m$ a squarefree integer
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 $K(v)$