Algebra: Volume I: Fields and Galois Theory by Falko Lorenz

By Falko Lorenz

From Math stories: "This is a captivating textbook, introducing the reader to the classical elements of algebra. The exposition is admirably transparent and lucidly written with basically minimum must haves from linear algebra. the hot options are, no less than within the first a part of the publication, outlined within the framework of the improvement of rigorously chosen difficulties. therefore, for example, the transformation of the classical geometrical difficulties on structures with ruler and compass of their algebraic surroundings within the first bankruptcy introduces the reader spontaneously to such primary algebraic notions as box extension, the measure of an extension, etc... The ebook ends with an appendix containing workouts and notes at the earlier elements of the publication. even if, short old reviews and proposals for extra examining also are scattered in the course of the text."

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A) Any field K has exactly one prime field as a subfield. ) (b) Any prime field K is isomorphic either to ‫ ޑ‬or to some ‫= ޚ‬p ‫ ޚ‬for p prime (depending on whether char K D 0 or char K D p > 0). F10. Proof. (a) The intersection of all subfields of K is a subfield of K. It is the smallest subfield of K, hence a prime field. (b) Let K be any field and K0 its prime field. Clearly, ‫ޚ‬K Â K0 . Now, in case A above, ‫ޚ‬K is itself already a subfield of K, so K0 D ‫ޚ‬K ' ‫= ޚ‬p ‫ ޚ‬. In case B we have K0 D Frac ‫ޚ‬K ' Frac ‫ ޚ‬D ‫ ޑ‬.

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.

We say that a 2 R can be decomposed into irreducible factors if it has an expression of the form (17) aD" 1 2 ::: r with " 2 R and each i irreducible: (Here we allow r D 0, in which case (17) is to be read as saying that a D "1 D "). An integral domain where every a ¤ 0 has a decomposition into irreducible factors is called a factorization domain. We say that a has a unique decomposition into irreducible factors if it has a decomposition into irreducible factors and the following uniqueness condition holds: If in addition to (17) we have another such decomposition (18) a D "0 0 0 1 2 ::: 0 r0 ; then r 0 D r and, after a permutation, i0 D O i for 1 Ä i Ä r .

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