By Reinhold Baer
Natural AND utilized MATHEMATICSA sequence of Monographs and TextbooksEdited byPaul A. Smith and Samuel Eilenberg Columbia collage, New YorkIn this publication we intend to set up the basic structural id of projective geometry and linear algebra. It has, after all, lengthy been discovered that those disciplines are exact. The proof substantiating this assertion is contained in a few theorems exhibiting that sure geometrical techniques might be represented in algebraic model. even if, it is extremely tough to find those basic lifestyles theorems within the literature inspite of their significance and nice usefulness. The middle of our dialogue will therefore be shaped by way of theorems of simply this kind. those are interested by the illustration of projective geometries through linear manifolds, of projectivities by way of semi-linear adjustments, of collineations by way of linear changes and of dualities by way of semi-bilinear types. those theorems will lead us to a reconstruction of the geometry which was once the start line of our discourse inside of such (apparently) merely algebraic constructions because the endomorphism ring of the underlying linear manifold or the entire linear team.
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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.