Algebraically closed group information

closed group. Every algebraically closed group is simple. No algebraically closed group is finitely generated. An algebraically closed group cannot be recursively...

their center (an algebraic torus) with a semisimple group. The latter are classified over algebraically closed fields via their Lie algebra. The classification...

determinant. Then a linear algebraic group G over an algebraically closed field k is a subgroup G(k) of the abstract group GL(n,k) for some natural number...

classification of reductive groups is the same over any algebraically closed field. In particular, the simple algebraic groups are classified by Dynkin diagrams...

Lie algebras have a very elegant classification, in stark contrast to solvable Lie algebras. Semisimple Lie algebras over an algebraically closed field...

If R is a ring or an algebra over a field, and S is a subset of R, then the centralizer of S is exactly as defined for groups, with R in the place of...

be embedded in the algebraically closed group G∗{\displaystyle G^{*}} then given another algebraically closed group H∗{\displaystyle H^{*}}, we can ask...

{\displaystyle K} is pseudo algebraically closed if it satisfies certain properties which hold for algebraically closed fields. The concept was introduced...

problem of classifying the simple Lie algebras. The simple Lie algebras of finite dimension over an algebraically closed field F of characteristic zero were...

finitely generated group is solvable if and only if the group can be embedded in every algebraically closed group. The rank of a group is often defined...

more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields,...

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...

{\displaystyle V} form an associative division algebra over the underlying field F. If F is algebraically closed, the only equivariant endomorphisms of an...

also characterized, among algebraic subgroups, by the condition that G/P is a complete variety. Working over algebraically closed fields, the Borel subgroups...

not algebraically closed. Classical algebraic varieties are the quasiprojective integral separated finite type schemes over an algebraically closed field...

semisimple elements. Over an algebraically closed field, a toral subalgebra is automatically abelian. Thus, over an algebraically closed field of characteristic...

Algebraic groups over algebraically closed fields have finite Morley rank, equal to their dimension as algebraic sets. Sela (2006) showed that free groups, and...

splittings are conjugate; thus split Lie algebras are of most interest for non-algebraically closed fields. Split Lie algebras are of interest both because they...

is that the algebraically closed groups are so wild that none of them has a recursive presentation. The oldest result relating algebraic structure to...

a = b2 or a = −b2. F is not algebraically closed, but its algebraic closure is a finite extension. F is not algebraically closed but the field extension F...

foundational results concerning Lie groups can be developed completely algebraically, giving rise to the theory of algebraic groups defined over an arbitrary field...

Because the theory of algebraically closed fields of characteristic zero is complete, a theory valid for a special algebraically closed field of characteristic...

division algebra A over a field K: dim A = 1 if K is algebraically closed, dim A = 1, 2, 4 or 8 if K is real closed, and If K is neither algebraically nor...

In algebraic geometry, an affine algebraic set is the set of the common zeros over an algebraically closed field k of some family of polynomials in the...

typically use algebraically founded homotopy continuation, with a base field of the complex numbers. Algebraic equation Computational group theory Dolotin...