Algebraic closure

In mathematics, particularly abstract algebra, an algebraic closure of a field K is an algebraic extension of K that is algebraically closed. It is one of many closures in mathematics.

Using Zorn's lemma, it can be shown that every field has an algebraic closure, and that the algebraic closure of a field K is unique up to an isomorphism that fixes every member of K. Because of this essential uniqueness, we often speak of the algebraic closure of K, rather than an algebraic closure of K.

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 over K form an algebraic closure of K.

The algebraic closure of a field K has the same cardinality as K if K is infinite, and is countably infinite if K is finite.

Let ${\displaystyle S=\{f_{\lambda }|\lambda \in \Lambda \}}$ be the set of all monic irreducible polynomials in K[x]. For each ${\displaystyle f_{\lambda }\in S}$, introduce new variables ${\displaystyle u_{\lambda ,1},\ldots ,u_{\lambda ,d}}$ where ${\displaystyle d={\rm {degree}}(f_{\lambda })}$. Let R be the polynomial ring over K generated by ${\displaystyle u_{\lambda ,i}}$ for all ${\displaystyle \lambda \in \Lambda }$ and all ${\displaystyle i\leq {\rm {degree}}(f_{\lambda })}$. Write

...
Wikipedia