Bounds in the theory of polynomial rings over fields. A nonstandard approach. (English) Zbl 0539.13011

In this paper nonstandard methods are used to show the existence of bounds for polynomial rings over fields of which the following are typical: Let \(f,f_ 1,...,f_ m\in K[X_ 1,...,X_ n]=K[X]\) of degree \(\leq d\), K a field. (a) \(f\in(f_ 1,...,f_ m)\) iff \(f=\Sigma f_ ih_ i\) for certain \(h_ i\in K[X]\) of degree \(\leq A\); (b) \((f_ 1,...,f_ m)\) is prime or contains 1 iff \[ gh\in(f_ 1,...,f_ m)\Rightarrow g\in(f_ 1,...,f_ m)\quad or\quad h\in(f_ 1,...,f_ m) \] holds for all g, \(h\in K[X]\) of degree \(\leq P\). - Here A and P are bounds depending only on (n,d) (not on K, m, or the polynomials in question).
The results are proved by considering the ring extension \(K[X]\subset K[X]_{int}\), K an internal field in some nonstandard structure, \(K[X]_{int}\) the ring of internal polynomials in \(X_ 1,...,X_ n\) over K. (a) follows from the faithful flatness of this extension, (b) from the results that prime ideals of K[X] remain prime when extended to \(K[X]_{int}\). - By concentrating on the existence of bounds (rather than their construction) many complications of the original constructive approach are avoided.


13F20 Polynomial rings and ideals; rings of integer-valued polynomials
12E05 Polynomials in general fields (irreducibility, etc.)
03H99 Nonstandard models
13B25 Polynomials over commutative rings
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