Generating subfields

Boolean FIP ring extensions

“…Let g(X) = (X − x)g ′ (X), where g ′ (X) ∈ L[X] is a product of some of the f α (X). A necessary and sufficient condition in order that there is K ∈ [k, L] such that g = f K is gotten for k = Q in [35,Remark 6], a result without proof that we supply for an arbitrary field k.…”

Functional Decomposition Using Principal Subfields

“…Let K/k be a separable field extension of finite degree n. A field L is said to be a subfield of K/k if k ⊆ L ⊆ K. It is well known that the number of subfields of K/k is not polynomially bounded in general. However, we have the following remarkable result from [19]:…”

“…. , L r are called principal subfields of the extension K/k and can be obtained as the kernel of some application (see [19]). Instead of directly searching for all subfields of a field extension, which leads to an exponential time complexity, principal subfields allow us to search for a specific set of r ≤ n subfields, a polynomial time task.…”

“…In this work, we try to improve the nonpolynomial part of the complexity. In order to achieve this, we make use of the so-called principal subfields, as defined in [19].…”

The complexity of computing all subfields of an algebraic number field

“…Since K/k is separable, g is separable as well. The following set is a subfield of K (see [16,Section 2])…”