query: Let $F/K$ be a field extension and $a\in F$ such that $[K(a):K]$ is odd
positive_passages.docid: 22430
positive_passages.text: Equal simple field extensions?
negative_passages.docid: 133561
negative_passages.text: show that if $K$ is a field then $K[x]$ is principal
negative_passages.docid: 121301
negative_passages.text: What are the roots of an irreducible $p\in F[X]$ in the field extension $F[X]/\langle p\rangle$?
negative_passages.docid: 50668
negative_passages.text: Show that $\mathbb{Q}(\sqrt2, \sqrt[3]2)$ is a primitive field extension of $\mathbb{Q}$.
negative_passages.docid: 142707
negative_passages.text: Intermediate fields of a finite field extension that is not separable
negative_passages.docid: 69935
negative_passages.text: Field extensions of finite degree and primitive elements
negative_passages.docid: 127796
negative_passages.text: Degree of field extension $F(x) / F(x^2 + 1 / x^2)$
negative_passages.docid: 127002
negative_passages.text: Prove that for a nonzero element $a$ of a field $F$ with $q$ elements $a^{-1} = a^{q-2}$
negative_passages.docid: 11698
negative_passages.text: $M /K \land L /K$ algebraic $\implies ML/K$ algebraic?
negative_passages.docid: 145594
negative_passages.text: Splitting of prime ideals in algebraic extensions
negative_passages.docid: 134619
negative_passages.text: Galois extension with $X^4-2aX^2+b\in\mathbb Q[X]$
negative_passages.docid: 158924
negative_passages.text: Example of composition of two normal field extensions which is not normal.
negative_passages.docid: 42652
negative_passages.text: Must $k$-subalgebra of $k[x]$ be finitely generated?
negative_passages.docid: 56551
negative_passages.text: Multiplicative group of field extension over base field
negative_passages.docid: 57283
negative_passages.text: Extending Homomorphism into Algebraically Closed Field
negative_passages.docid: 145140
negative_passages.text: Subfield of $\mathbb{Q}(\sqrt[n]{a})$
negative_passages.docid: 111129
negative_passages.text: Show that $[K:F]_s = [K:L]_s [L:F]_s$ and $[K:F]_i = [K:L]_i [L:F]_i$.