Algebra, Volume 1: Fields and Galois Theory (Universitext) by Falko Lorenz

By Falko Lorenz

The current textbook is a full of life, problem-oriented and thoroughly written creation to classical glossy algebra. the writer leads the reader via attention-grabbing material, whereas assuming simply the history supplied through a primary path in linear algebra.

The first quantity specializes in box extensions. Galois thought and its purposes are handled extra completely than in such a lot texts. It additionally covers uncomplicated purposes to quantity thought, ring extensions and algebraic geometry.

The major concentration of the second one quantity is on extra constitution of fields and comparable themes. a lot fabric no longer frequently lined in textbooks looks right here, together with genuine fields and quadratic kinds, diophantine dimensions of a box, the calculus of Witt vectors, the Schur crew of a box, and native classification box theory.

Both volumes include quite a few workouts and will be used as a textbook for complex undergraduate scholars.

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Ad C bc/=bd ; Œa=b Œc=d  D Œac=bd : Checking that these operations are well defined is left to the reader. It is easy to see that with these operations K becomes a commutative ring with unity; the zero element is Œ0=1 and the unity is Œ1=1. The map à W R ! a/ D Œa=1 is a homomorphism. By definition, Œa=b D 0 D Œ0=1 if and only if a D 0. In particular, à is injective. In addition, every Œa=b ¤ 0 in K has a multiplicative inverse, namely Œb=a. Therefore K is a field. ˜ The classical example of the construction above is the field of rational numbers ‫ ޑ‬D Frac ‫ ޚ‬: Other key examples arise as follows: Definition.

But none of this would help if we could not prove the existence of interesting principal ideal domains. . F5. The ring ‫ ޚ‬of integers is a principal ideal domain. 36 4 Fundamentals of Divisibility Proof. 0/. Among all nonzero elements of I , let a be one with smallest absolute value jaj. a/. a/ Â I . Now let b 2 I . By considering division with remainder we see that there exist q; r 2 ‫ ޚ‬such that b D qa C r and jr j < jaj (we can even demand that 0 Ä r < jaj or alternatively that 12 jaj < r Ä 12 jaj/.

Since f D qg C r with deg r < deg g we first get f j r h, and since the degree of g is minimal and less than that of f we next get r D 0. Because f is irreducible it follows that g is a unit — a contradiction. ˜ Theorem 2 was first formulated by Simon Stevin in 1585; the analogous statement for the ring ‫ ޚ‬is already in the works of Euclid (ca. 330). F5. Kf D KŒX =f is a field if and only if f is irreducible in KŒX . Proof. Let Kf be a field. ˛/ D 0. Because of (20), either f2 or f1 lies in K, so f is irreducible.

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