splitting field of a finite set of polynomials
If is the ideal generated by in , since is irreducible, is a maximal ideal of , and consequently is a field.
We can construct a canonical monomorphism from to . By tracking back the field operation on , can be extended to an isomorphism from an extension field of to .
We show that is a root of .
If we write then implies:
which means that .∎
If is a field extension of then the nonconstant polynomials split in iff the polynomial splits in . Now the proof easily follows from the above lemma. ∎
|Title||splitting field of a finite set of polynomials|
|Date of creation||2013-03-22 16:53:09|
|Last modified on||2013-03-22 16:53:09|
|Last modified by||polarbear (3475)|