Dirichlet’s Theorem for Polynomial Rings
نویسنده
چکیده
We prove the following form of Dirichlet’s theorem for polynomial rings in one indeterminate over a pseudo algebraically closed field F . For all relatively prime polynomials a(X), b(X) ∈ F [X] and for every sufficiently large integer n there exist infinitely many polynomials c(X) ∈ F [X] such that a(X) + b(X)c(X) is irreducible of degree n, provided that F has a separable extension of degree n.
منابع مشابه
Euclidean Proofs of Dirichlet’s Theorem
Euclid’s proof of the infinitude of the primes is a paragon of simplicity: given a finite list of primes, multiply them together and add one. The resulting number, say N , is not divisible by any prime on the list, so any prime factor of N is a new prime. Some special cases of Dirichlet’s theorem admit a simple proof following Euclid’s model, such as the case of 1 mod 4 or 5 mod 6. (We mean by ...
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