Download PDF by Fulton W.: Algebraic curves

By Fulton W.

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1 ................ x1 a) Show that the complement Λ of a monoideal in a monoid is characterized by the following property: if γ ∈ Λ and γ | γ , then γ ∈ Λ . 8. Show that ∆(I) is finitely cogenerated and find a minimal set of cogenerators.

X31 x2 • 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .......................

Fm ), as claimed. Now it is time to move directly to the heart of this section. We want to prove that polynomial rings over fields are factorial. The next lemma is the key to this proof. 9. Let R be a factorial domain and f ∈ R[x] \ {0}. A greatest common divisor of the coefficients of f is called a content of f . As before, we usually speak of the content of f and denote it by cont(f ). If cont(f ) = 1 , we say that f is primitive. 10. (Gauß’s Lemma) Let R be a factorial domain, and let f, g ∈ R[x] be non-zero polynomials.

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Algebraic curves by Fulton W.


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