# Algebra by Serge Lang

By Serge Lang

This publication is meant as a simple textual content for a one-year path in Algebra on the graduate point, or as an invaluable reference for mathematicians and pros who use higher-level algebra. It effectively addresses the fundamental thoughts of algebra. For the revised 3rd version, the writer has further workouts and made quite a few corrections to the text.

Comments on Serge Lang's Algebra:

"Lang's Algebra replaced the way in which graduate algebra is taught, protecting classical subject matters yet introducing language and methods of pondering from class conception and homological algebra. It has affected all next graduate-level algebra books."

-April 1999 Notices of the AMS, asserting that the writer was once presented the Leroy P. Steele Prize for Mathematical Exposition for his many arithmetic books.

"The writer has a magnificent knack for providing the $64000 and fascinating rules of algebra in exactly the "right" approach, and he by no means will get slowed down within the dry formalism which pervades a few components of algebra."

-MathSciNet's assessment of the 1st version

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**Example text**

Free on one generator, this proves that B is free. (When A is not finitely generated, one can use a simi lar transfinite argument. ) We a lso observe that our proof shows that there exists at least one basis of B whose cardinality is ~ n. We shalJ therefore be finished when we prove the last statement, that an y two bases of B ha ve the same cardinality. Let S be one basis, with a finite number of elements m. Let T be another basis, and suppose that T has at least r elements. It will suffice to prove that r ~ m (one 42 GROUPS I, §8 can then use symmetry).

If a generates G, then G is cyclic . We also say that a has infinite period. 2. The kernel is not trivial. Let d be the smallest positive integer in the kernel. Then d is called the period of a. If m is an integer such that am = e then m = ds for some integer s. We observe that the elements e, a, . . , ad - \ are 24 GROUPS I, §4 distinct. Indeed , if ar = as with 0 ~ r, s ~ d - 1, and say r ~ s, then a"? = e . Since 0 ~ s - r < d we must have s - r = O. The cyclic subgroup generated by a has order d.

3. Example from linear algebra. We assume the reader knows basic notions of linear algebra . Let k be a field and let V be a vector space over k . Let G = GL(V) be the group of linear automorphisms of V . For A E G and v E V, the map (A , v) ~ Av defines an operation of G on V. Of course , G is a subgroup of the group of permutations Perm(V). Similarly , let V = k" be the vector space of (vertical) n-tuples of elements of k, and let G be the group of invertible n x n matrices with components in k .