# Basic Theory of Algebraic Groups and Lie Algebras by G. P. Hochschild

By G. P. Hochschild

The concept of algebraic teams effects from the interplay of assorted uncomplicated strategies from box idea, multilinear algebra, commutative ring concept, algebraic geometry and normal algebraic illustration idea of teams and Lie algebras. it really is therefore an preferably appropriate framework for displaying easy algebra in motion. to do this is the central drawback of this article. for this reason, its emphasis is on constructing the most important common mathematical instruments used for gaining keep an eye on over algebraic teams, instead of on securing the ultimate definitive effects, similar to the class of the straightforward teams and their irreducible representations. within the comparable spirit, this exposition has been made completely self-contained; no distinctive wisdom past the standard common fabric of the 1st one or years of graduate examine in algebra is pre­ meant. The bankruptcy headings might be adequate indication of the content material and organization of this e-book. every one bankruptcy starts off with a quick statement of its effects and ends with a number of notes starting from supplementary effects, amplifications of proofs, examples and counter-examples via routines to references. The references are meant to be purely feedback for supplementary interpreting or symptoms of unique assets, specifically in instances the place those may not be the predicted ones. Algebraic workforce concept has reached a kingdom of adulthood and perfection the place it might now not be essential to re-iterate an account of its genesis. Of the cloth to be offered right here, together with a lot of the fundamental help, the main component is because of Claude Chevalley.

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As a first application, this section establishes the expected relation between the dimensions of G, Hand G/H, where H is a normal algebraic subgroup of the algebraic group G. 1. 1. , ... , ur) be a set of independent variables over K. Let S be afinite group offield automorphisms of K(Ul" .. ,ur) that stabilizes K as well as the multiplicative group generated by the u;'s. Suppose that the representation of Son K is injective. Then K(Ub ... ,ur)S is contained in a finitely generated purely transcendental extension field of K S• PROOF.

Ur ). First, we show that K is separably algebraic over F(X). Let t be the smallest index 2: r such that K is separably algebraic over F(Ul' . , Ut). We shall obtain a contradiction from the assumption that t is strictly greater than r. 2 F(U1, ... 2, it follows that F(UI,' .. 3, the element Ut is therefore not separably algebraic over F(u 1 , ••• , ut - 1 ). Let us first deal with the case where U t is algebraic over F(u 1,· .. , U t -1)' In this case, F must be of non-zero characteristic p, and the monic minimum polynomial,fsay, for Ut relative to F(UI,"" ut - 1) must satisfy f(x) = g(x P), where x is an auxiliary variable, and g is a polynomial with coefficients in F(U1'" .

Let f denote the monic minimum polynomial for s relative to F(X), but view f as a polynomial with coefficients in F(X') via the isomorphism just mentioned. Since s is separable over F(X), the polynomialfis the product of a set of mutually distinct monic irreducible factors with coefficients in L(X'). It follows from this that K ® F(X) L(X') is a direct sum of fields, one for each irreducible factor off Therefore, our nilpotent element u must be 0. Now suppose that the condition of the theorem is satisfied.