query: Equivalent criterion for algebraic independence over $\mathbb{Q}$.
positive_passages.docid: 145039
positive_passages.text: Algebraic independence over $\overline{\mathbb Q}$ and over $\mathbb Q$
negative_passages.docid: 1117
negative_passages.text: Is there a bijection between $(0,1)$ and $\mathbb{R}$ that preserves rationality?
negative_passages.docid: 1914
negative_passages.text: General way to determine $\mathbb{Q}(\gamma) = \mathbb{Q}(\alpha,\beta)$ given $\alpha$ and $\beta$
negative_passages.docid: 68430
negative_passages.text: Proving the countability of algebraic numbers
negative_passages.docid: 155828
negative_passages.text: How do I show that $A$ and $A^T$ are similar, given that the minimal polynomial of $A$ splits over $\mathbb{F}$?
negative_passages.docid: 95928
negative_passages.text: Independence and orthogonality
negative_passages.docid: 111248
negative_passages.text: The Uniqueness of a Coset of $R[x]/\langle f\rangle$ where $f$ is a Polynomial of Degree $d$ in $R[x]$
negative_passages.docid: 103552
negative_passages.text: The minimal polynomial of a primitive $p^{m}$-th root of unity over $\mathbb{Q}_p$
negative_passages.docid: 99884
negative_passages.text: Are $\mathbb{Q}_p$ and $\mathbb{Q}_q$ homeomorphic?
negative_passages.docid: 69046
negative_passages.text: Proof of $\mathbb{Q}$ is not cyclic
negative_passages.docid: 11698
negative_passages.text: $M /K \land L /K$ algebraic $\implies ML/K$ algebraic?
negative_passages.docid: 1060
negative_passages.text: Minimal polynomial of intermediate extensions under Galois extensions.
negative_passages.docid: 14270
negative_passages.text: Polynomial divisibility is unchanged by coeff. field extension
negative_passages.docid: 100257
negative_passages.text: Converse to Hilbert basis theorem
negative_passages.docid: 40858
negative_passages.text: Multiplicative nature of the separability degree
negative_passages.docid: 83173
negative_passages.text: Prove linear combinations of logarithms of primes over $\mathbb{Q}$ is independent
negative_passages.docid: 78561
negative_passages.text: Algebraic field extensions: Why $k(\alpha)=k[\alpha]$.