By Falko Lorenz
From Math stories: "This is a captivating textbook, introducing the reader to the classical components of algebra. The exposition is admirably transparent and lucidly written with in simple terms minimum necessities from linear algebra. the hot techniques are, no less than within the first a part of the ebook, outlined within the framework of the improvement of conscientiously chosen difficulties. hence, for example, the transformation of the classical geometrical difficulties on structures with ruler and compass of their algebraic environment within the first bankruptcy introduces the reader spontaneously to such primary algebraic notions as box extension, the measure of an extension, etc... The booklet ends with an appendix containing routines and notes at the earlier components of the ebook. although, short historic reviews and proposals for extra interpreting also are scattered during the text."
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Extra info for Algebra. Fields and Galois Theory
Here is the best known application of this approach: F10 (Eisenstein irreducibility criterion). X / D aX n C an 1X n 1 C C a1 X C a0 be a primitive polynomial in RŒX . If there exists a prime of R such that (i) - a, (ii) j ai for 0 Ä i Ä n 1, and (iii) 2 - a0 , then f is irreducible in RŒX . If R is a UFD, f is also irreducible in KŒX , where K is the fraction ﬁeld of R. Proof. Since was assumed prime in R, the quotient R WD R= is an integral domain. We now work as in the proof of F9. Suppose f has a nontrivial factorization in RŒX : f D gh; with r D deg g 1 and s D deg h 1: Taking the image in RŒX leads to, as before, f D gh; with deg g D r and deg h D s: To be sure, f is not irreducible, but because of condition (ii) it has the form f D aX n , so that in RŒX the equation aX n D gh (14) holds.
Now keeping in mind Chapter 4, F11, the proof is complete. ˜ F6. Let R be a UFD with K D Frac R, and let g 2 RŒX be nonconstant. If g is irreducible in RŒX , it is irreducible in KŒX . Irreducibility criteria and applications 49 Proof. By Theorem 1 we have (with ᏼ2 as in the theorem’s statement) g D "f with " 2 R and f 2 ᏼ2 : Thus g is irreducible in KŒX . ˜ Conversely, if g 2 RŒX is irreducible in KŒX , then g is irreducible in RŒX if and only if g is primitive. F7 (Gauss’s Lemma). Let R be a UFD and K D Frac R.
D /, then d is a gcd of a; b. ) Deﬁnition 3. An integral domain R is called a principal ideal domain, or PID, if every ideal of R is a principal ideal. F4. In a principal ideal domain R any tuple of elements a1 ; : : : ; an of R has a gcd. If d is a gcd of a1 ; : : : ; an , it can be represented as (14) d D x1 a 1 C C xn an for appropriate xi 2 R. Proof. d /: This means d is a gcd of the ai , by the argument preceding Deﬁnition 3; moreover, d clearly has a representation of the desired from. d /, by F2 and (11).