Galois extension

Target entity: Galois extension
Context triple: [Die Theorie der algebraischen Zahlkörper, usesConcept, Galois extension]

• A. Galois group
  A Galois group is the group of field automorphisms of a field extension that captures the symmetries of its algebraic equations and underpins much of modern algebra and number theory.
• B. Galois theory
  Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
• C. Kronecker–Weber theorem
  The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
• D. Galois
  Galois is a French surname most famously associated with Évariste Galois, the pioneering 19th-century mathematician who founded group theory and laid the groundwork for modern abstract algebra.
• E. Kummer theory
  Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
• F. None of above. chosen
• G. Unsure - the case is ambiguous/there is not enough information to decide.

Target entity: Galois extension
Target entity description: A Galois extension is a field extension that is both normal and separable, characterized by a well-structured group of automorphisms known as its Galois group.

• A. Galois group
  A Galois group is the group of field automorphisms of a field extension that captures the symmetries of its algebraic equations and underpins much of modern algebra and number theory.
• B. Galois theory
  Galois theory is a branch of abstract algebra that studies field extensions and polynomial equations through the structure of their associated symmetry groups.
• C. Kronecker–Weber theorem
  The Kronecker–Weber theorem is a fundamental result in algebraic number theory stating that every finite abelian extension of the rational numbers is contained in a cyclotomic field generated by roots of unity.
• D. Galois
  Galois is a French surname most famously associated with Évariste Galois, the pioneering 19th-century mathematician who founded group theory and laid the groundwork for modern abstract algebra.
• E. Kummer theory
  Kummer theory is a branch of algebraic number theory that studies abelian extensions of fields, especially cyclotomic and radical extensions, using properties of roots of unity and ideal class groups.
• F. None of above. chosen