Selections from Modern Abstract Algebra.

Concrete proofs with abstract objects in modern algebra

p. 2 —————— The computer algebra system D5 ————– Classical Theorem. Any field K is contained in an algebraically closed field. Recursive counter example. It is possible to construct a computable field K ⊇ Q in which there is no (recursive) algorithm for factoring polynomials X − p (p a prime). First classical step. Given any polynomial f of degree d ≥ 1 in K[X] there exists a field L ⊇ K where ...

Modern Algebra

Algebra is the study of mathematical manipulations that preserve something (like equality –when solving equations). The areas in which Algebra finds application are quite diverse, from Ancient Greek Geometry through to Modern Information Protection and Security (error correcting codes, data compression, and cryptography). This course begins with topics that should be familiar (such as rulerand-...

Abstract Algebra

Algebra

Modern Algebra with Applications

18.703 Modern Algebra, Cosets

Consider the group of integers Z under addition. Let H be the subgroup of even integers. Notice that if you take the elements of H and add one, then you get all the odd elements of Z. In fact if you take the elements of H and add any odd integer, then you get all the odd elements. On the other hand, every element of Z is either odd or even, and certainly not both (by convention zero is even and...