Algebra began when quadratic equations were solved by al-Khowarizmi. Its next step was the solution of third and fourth degree equations, published by Cardano in [1545]. Equations of degree 5, however, resisted all efforts at similar solutions, until Abel [1824] and Galois [1830] proved that no such solution exists. Abel’s solution did not hold the germs of future progress, but Galois’s ideas initiated the theory that now bears his name, even though Galois himself lacked a clear definition of fields. The modern version has remained virtually unchanged since Artin’s lectures in the 1920s. Galois theory provides a one-to-one correspondence betweenintermediate fields K ⊆ F ⊆ E of suitable extensions and subgroups of their groups of K-automorphisms.
This allows group theory to apply to fields. For instance, a polynomial equation is solvable by radicals if and only if the corresponding group is solvable Splitting Fields The splitting field of a set of polynomials is the field generated by their roots in some algebraic closure. This section contains basic properties of splitting fields, and the determination of all finite fields.that every polynomial with coefficients in a field K has a root in some field extension of K . A polynomial splits in an extension when it has all its roots in that extension:
Definition. A polynomial f ∈ K[X] splits in a field extension E of K when it has a factorization f (X) = a (X − α1)(X − α2) · · · (X − αn) in E[X] . In the above, a ∈ K is the leading coefficient of f , n is the degree of f , and α1, . . ., αn ∈ E are the (not necessarily distinct) roots of f in E . For example,
every polynomial f ∈ K[X] splits in the algebraic closure K of K . Definition. Let K be a field. A splitting field over K of a polynomialf ∈ K[X] is a field extension E of K such that f splits in E and E is generated over K by the roots of f . A splitting field over K of a set S ⊆ K[X] of polynomials is a field extension E of K such that every f ∈ S splits in E and E is generated over K by the roots of all f ∈ S. In particular, splitting fields are algebraic extensions, by 3.3. Every set S ⊆ K[X] of polynomials has a splitting field, which is generated over K by the roots
of all f ∈ S in K , and which we show is unique up to K-isomorphism.
This allows group theory to apply to fields. For instance, a polynomial equation is solvable by radicals if and only if the corresponding group is solvable Splitting Fields The splitting field of a set of polynomials is the field generated by their roots in some algebraic closure. This section contains basic properties of splitting fields, and the determination of all finite fields.that every polynomial with coefficients in a field K has a root in some field extension of K . A polynomial splits in an extension when it has all its roots in that extension:
Definition. A polynomial f ∈ K[X] splits in a field extension E of K when it has a factorization f (X) = a (X − α1)(X − α2) · · · (X − αn) in E[X] . In the above, a ∈ K is the leading coefficient of f , n is the degree of f , and α1, . . ., αn ∈ E are the (not necessarily distinct) roots of f in E . For example,
every polynomial f ∈ K[X] splits in the algebraic closure K of K . Definition. Let K be a field. A splitting field over K of a polynomialf ∈ K[X] is a field extension E of K such that f splits in E and E is generated over K by the roots of f . A splitting field over K of a set S ⊆ K[X] of polynomials is a field extension E of K such that every f ∈ S splits in E and E is generated over K by the roots of all f ∈ S. In particular, splitting fields are algebraic extensions, by 3.3. Every set S ⊆ K[X] of polynomials has a splitting field, which is generated over K by the roots
of all f ∈ S in K , and which we show is unique up to K-isomorphism.
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