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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Extra info for Algebra. Fields and Galois Theory
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.