Get Algebraic Closure essential facts below. View Videos or join the Algebraic Closure discussion. Add Algebraic Closure to your PopFlock.com topic list for future reference or share this resource on social media.
Notion that any algebraic equation with coefficients in a structure has a solution in the structure
The algebraic closure of a field K can be thought of as the largest algebraic extension of K.
To see this, note that if L is any algebraic extension of K, then the algebraic closure of L is also an algebraic closure of K, and so L is contained within the algebraic closure of K.
The algebraic closure of K is also the smallest algebraically closed field containing K,
because if M is any algebraically closed field containing K, then the elements of M that are algebraic overK form an algebraic closure of K.
There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers, e.g. the algebraic closure of Q(?).
Existence of an algebraic closure and splitting fields
Let be the set of all monic irreducible polynomials in K[x].
For each , introduce new variables where .
Let R be the polynomial ring over K generated by for all and all . Write
Let I be the ideal in R generated by the . Since I is strictly smaller than R,
Zorn's lemma implies that there exists a maximal ideal M in R that contains I.
The field K1=R/M has the property that every polynomial with coefficients in K splits as the product of and hence has all roots in K1. In the same way, an extension K2 of K1 can be constructed, etc. The union of all these extensions is the algebraic closure of K, because any polynomial with coefficients in this new field has its coefficients in some Kn with sufficiently large n, and then its roots are in Kn+1, and hence in the union itself.
It can be shown along the same lines that for any subset S of K[x], there exists a splitting field of S over K.
An algebraic closure Kalg of K contains a unique separable extensionKsep of K containing all (algebraic) separable extensions of K within Kalg. This subextension is called a separable closure of K. Since a separable extension of a separable extension is again separable, there are no finite separable extensions of Ksep, of degree > 1. Saying this another way, K is contained in a separably-closed algebraic extension field. It is unique (up to isomorphism).
The separable closure is the full algebraic closure if and only if K is a perfect field. For example, if K is a field of characteristic p and if X is transcendental over K, is a non-separable algebraic field extension.