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On a Special Case of Hilbert's Irreducibility Theorem

✍ Scribed by Marius Cavachi

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
89 KB
Volume
82
Category
Article
ISSN
0022-314X

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✦ Synopsis

We prove that if K is a finite extension of Q, P is the set of prime numbers in Z that remain prime in the ring R of integers of K, f, g # K[X] with deg g>deg f and f, g are relatively prime, then f +pg is reducible in K[X] for at most a finite number of primes p # P. We then extend this property to polynomials in more than one indeterminate. These results are related to Hilbert's irreducibility theorem.

2000 Academic Press

Throughout this paper, an algebraic number field K is such that the extension Q K is normal. We shall denote by R the ring of algebraic integers of K. It is well-known that the set P of prime numbers in Z that remain prime in R is infinite (see [2, p. 136]).

If f, g # K[X] are relatively prime, by Hilbert's irreducibility theorem, the irreducible polynomial f (X)+Yg(X) # K[X, Y] remains irreducible in K[X] for infinitely many Y=n # Z. We shall make this property more precise in our particular context.

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