query: Characterizing quadratic number fields that are subfields of cyclic quartic number fields
positive_passages.docid: 24168
positive_passages.text: A characterisation of quadratic extensions contained in cyclic extensions of degree 4
negative_passages.docid: 86515
negative_passages.text: Characterization of integers which has a $2$-adic square root
negative_passages.docid: 103654
negative_passages.text: A finite field extension that is not simple
negative_passages.docid: 80295
negative_passages.text: Field extensions and algebraic/transcendental elements
negative_passages.docid: 9520
negative_passages.text: Subfields of $\Bbb Q(\zeta_8)$
negative_passages.docid: 125568
negative_passages.text: Construction of $p^n$ field
negative_passages.docid: 151978
negative_passages.text: Splitting field of x^3-2 as a simple extension
negative_passages.docid: 115129
negative_passages.text: Constructing an explicit isomorphism between finite extensions of finite fields
negative_passages.docid: 56551
negative_passages.text: Multiplicative group of field extension over base field
negative_passages.docid: 136770
negative_passages.text: Any field that contains the integers contains the rationals as a subfield (Analysis proof)
negative_passages.docid: 98803
negative_passages.text: Square roots of integers and cyclotomic fields
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: 117789
negative_passages.text: Computing Galois Group of $\mathbb{Q}(\sqrt{2},\sqrt{3}):\mathbb{Q}$
negative_passages.docid: 79725
negative_passages.text: Fields of arbitrary cardinality
negative_passages.docid: 114152
negative_passages.text: Intermediate fields between $\mathbb{Z}_2 (\sqrt{x},\sqrt{y})$ and $\mathbb{Z}_2 (x,y)$
negative_passages.docid: 43155
negative_passages.text: Cyclotomic Polynomials and GCD
negative_passages.docid: 622
negative_passages.text: Galois Group of $\sqrt{2+\sqrt{2}}$ over $\mathbb{Q}$