Algebra. Fields and Galois Theory by Falko Lorenz

By Falko Lorenz

From Math reports: "This is an enthralling textbook, introducing the reader to the classical elements of algebra. The exposition is admirably transparent and lucidly written with merely minimum necessities from linear algebra. the hot techniques are, not less than within the first a part of the booklet, outlined within the framework of the improvement of rigorously chosen difficulties. hence, for example, the transformation of the classical geometrical difficulties on buildings with ruler and compass of their algebraic atmosphere within the first bankruptcy introduces the reader spontaneously to such basic algebraic notions as box extension, the measure of an extension, etc... The e-book ends with an appendix containing routines and notes at the prior elements of the booklet. in spite of the fact that, short historic reviews and recommendations for additional interpreting also are scattered in the course of the text."

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In many cases there is a canonical choice for ᏼ — for example, in R D ‫ ޚ‬the set of natural prime numbers stands out, and in the polynomial ring KŒX  over a field K we can take for ᏼ the set of all normalized prime polynomials. In any case we have: F10. Let R be a unique factorization domain and ᏼ a directory of primes of R. Every nonzero a 2 R possesses a unique representation of the form Y e ; (22) aD" 2ᏼ where " is a unit of R and the e are nonnegative integers with e D 0 for almost all 2 ᏼ (that is, all but finitely many 2 ᏼ).

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. Conversely, assume that f is irreducible. We already know that Kf is finitedimensional over K; keeping in mind Chapter 2, F2, we then just have to show that Kf is an integral domain.

For simplicity we set w D w . g1 h1 /. g1 h1 / > 0, that is, j g1 h1 . h1 / > 0. But this contradicts (9). ˜ 48 5 Prime Factorization in Polynomial Rings. Gauss’s Theorem Definition. A nonconstant polynomial f 2 RŒX  (that is, one whose degree is at least 1) is called primitive if the gcd of the coefficients of f is 1. Thus a normalized polynomial in RŒX  is trivially primitive. If R is a UFD, every nonconstant polynomial g 2 RŒX  can be represented as g D ag1 ; with a 2 R r f0g and g1 2 RŒX  primitive: Also, a is determined up to associatedness, being the gcd of the coefficients of g.

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