Field Extensions & Algebraic Closure

← Back to Domain-Specific Families

Abstractions about algebraically closed and pseudo-closed fields, extension degrees, minimal and separable polynomials, norm groups, and algebraic K-theory.

8 abstractions in this family — domain-specific abstractions that sit near one another in structural-signature space (k-means over structural-signature embeddings). Each is shown with its short description.

• Algebraically closed field — A field in which every nonconstant univariate polynomial with coefficients in that field has a root, equivalently splits completely into linear factors.
• Degree of a field extension — The vector-space dimension of an extension field over its base field.
• Milnor K-theory — The graded ring generated by nonzero field elements modulo Steinberg relations, linking symbols in algebraic K-theory with Galois cohomology.
• Minimal polynomial (field theory) — The unique monic polynomial of least positive degree over a base field having a given algebraic extension element as a root.
• Norm group — The subgroup of a local field’s multiplicative group consisting of field norms from a finite abelian extension.
• Pseudo algebraically closed field — A field in which every absolutely irreducible variety defined over the field has a rational point, imitating a key geometric consequence of algebraic closure without requiring every polynomial to split.
• Rupture field — A field extension generated by adjoining one root of a polynomial over the base field.
• Separable polynomial — A polynomial over a field whose roots in an algebraic closure are all distinct.