How is a splitting field constructed?

Adjoin a root of an irreducible factor by quotienting the polynomial ring by that factor, then repeat over the larger field until the polynomial factors into linear pieces. The resulting minimal field is the splitting field.

Why are splitting fields important for Galois theory?

A splitting field is exactly a normal extension, and when separable it is a Galois extension. Its Galois group permutes the roots of the polynomial, so the splitting field is where the symmetry analysis of the equation takes place.

Splitting Field

A splitting field of a polynomial is the smallest field extension over which the polynomial factors completely into linear factors, the natural arena in which all its roots live.

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Definition

A splitting field of a polynomial over a field is an extension generated by all roots of the polynomial in which it factors into linear factors, and which is minimal with that property.

Scope

This topic covers the construction and existence of splitting fields, their uniqueness up to isomorphism, normal extensions, the connection to algebraic closures, and the role of splitting fields as the Galois extensions in which a polynomial's roots and symmetries are studied.

Core questions

Why does every polynomial have a field in which it splits completely?
Is the splitting field of a polynomial unique?
How do splitting fields relate to normal extensions and algebraic closures?
Why are splitting fields the right setting for Galois theory?

Key theories

Existence and uniqueness of splitting fields: Every polynomial over a field has a splitting field, obtained by successively adjoining roots, and any two splitting fields of the same polynomial are isomorphic by an isomorphism fixing the base field.
Splitting fields and normality: A finite extension is normal exactly when it is the splitting field of some polynomial, equivalently when it contains all conjugates of each of its elements, which is one of the conditions defining a Galois extension.
Algebraic closure as a universal splitting field: An algebraic closure of a field is an extension in which every polynomial splits, and it is the union of the splitting fields of all polynomials, existing and being unique up to isomorphism for every field.

Clinical relevance

Splitting fields provide the concrete extensions on which Galois groups act, making them the foundation for computing Galois groups and for studying solvability of equations. The same construction yields algebraic closures and is used to build finite fields of every prime-power order.

History

Kronecker's method of adjoining roots by quotienting polynomial rings gives the construction of splitting fields, and Steinitz proved the existence and uniqueness of algebraic closures in his 1910 theory of abstract fields. These results put Galois's implicit use of root fields on rigorous footing.

Key figures

Leopold Kronecker
Ernst Steinitz
Évariste Galois

Seminal works

dummit2004
lang2002
hungerford1974

Frequently asked questions

How is a splitting field constructed?: Adjoin a root of an irreducible factor by quotienting the polynomial ring by that factor, then repeat over the larger field until the polynomial factors into linear pieces. The resulting minimal field is the splitting field.
Why are splitting fields important for Galois theory?: A splitting field is exactly a normal extension, and when separable it is a Galois extension. Its Galois group permutes the roots of the polynomial, so the splitting field is where the symmetry analysis of the equation takes place.