An Elimination Theory for Differential Algebra by A Seidenberg

By A Seidenberg

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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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