splitting field
A 'splitting field' is a fundamental concept in abstract algebra and Galois theory, crucial for understanding polynomial factorization and field extensions.
🇺🇸 US Voice:
🇬🇧 UK Voice:
Definition
C2Galois Theory
(technical, academic)The smallest field extension of a given field where a polynomial can be factored into linear terms.
Example
- The splitting field of the polynomial x^2 - 2 over the rationals is the field obtained by adjoining the square root of 2.
C2Ring Theory
(technical, academic)An extension field where every simple module of a finite-dimensional algebra remains simple.
Example
- In ring theory, identifying the splitting field helps in analyzing the structure of modules.
C2Character Theory
(technical, academic)A field over which a group representation includes every irreducible character of the group.
Example
- The splitting field of a character in representation theory simplifies the study of group actions.
Similar
Terms that have similar or relatively close meanings to "splitting field":
split endsplit potsplit keyfield upsplit upbroken fieldhigh-low splitdividing linesplit hairs