Dirichlet’s Theorem for Polynomial Rings

1. Dirichlet’s Theorem for Polynomial Rings Lior Bary-Soroker,School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel-Aviv University 1. The classical Dirichlet Theorem for primes in arithmetical progressions 3. Pseudo algebraically closed fields and Hilbert’s Irreducibility Theorem Hilbert’s Irreducibility Theorem (1891) states that given an irreducible polynomial f(t,x)with two variables over the rational numbers we can substitute t with (infinitely many) rational numberst0such that f(t0,x)remainsirreducible (as a polynomial in one variable). Johann Peter Gustav Lejeune Dirichlet, German mathematician, 1805-1859, who deeply influenced all areas of mathematics including: Number Theory, Analysis, Probability, Combinatorics, etc. David Hilbert, German mathematician, 1862-1943, one of the most influential and universal mathematicians of the 19th and early 20th centuries. In 1900 posed his celebrated 23 questions. The prime numbers are the building blocks of the integers. The classical Dirichlet Theorem (1837) asserts that in every arithmetical progression there are infinitely many primes, unless there is an obvious reason why not. 1(mod 4): 1 5 9 1317 21 25 29 33 3741 45… 2(mod 5): 2 7 12 17 22 27 32 37 42 47 52… 6(mod 22): 6 28 50 72 94 116 138 160 172 … A pseudo algebraically closed (PAC) field K is a field with the geometrical feature that every absolutely irreducible variety over it has a K-rational point. We show, using methods of Roquette, that these fields satisfy a weaker version of Hilbert’s Irreducibility Theorem: Theorem: Let K be a PAC field and let f(t,x) be an irreducible polynomial of degree n in x and with the symmetric group Sn as the Galois group over L(t) where L is an algebraic closure of K. Then if K has a separable extension of degree n, then there exist infinitely many t0in K such that f(t0 ,x) is irreducible over K. Three arithmetical progressions; the primes are colored with red. Note: In the last row, all the numbers are even and thus there are no primes. 2. An analog of Dirichlet Theorem for polynomial rings 4. The ramification determines the Galois group of a polynomial In this work we are interested in polynomials over a field. There is an analogy between polynomials over a field and the integer numbers. We can multiply and add polynomials, divide with remainder, etc… In particular there are ``prime polynomials” (or equivalently irreducible polynomials) and “arithmetical progressions.” Elements in the Galois group of an irreducible polynomial f(t,x)over K(t)can be computed at ramification points of a branch cover of the projective line C→P1 that this polynomials defines. We force ramification type (2,1,...,1) at one point and (e,1,...,1) at another point (for a good choice of e>n/2) to get: Proposition: Let K be an infinite field with algebraic closure L, let a(x), b(x) be relatively prime polynomials over K, and let n be a large integer. Then there exists c(x) such that f(t,x) = a(x) + t b(x) c(x) is of degree n in x and with Galois group Sn over L(t). Some elements in the arithmetical progression X2(mod x+1). Irreducible polynomials over the rational numbers are marked with red. The goal of this work is an analog of Dirichlet’s theorem: Main Theorem: Let K be a pseudo algebraically closed field (see section 3). Then, for relatively prime polynomials a(x), b(x) over K and sufficiently large integer n, there exist infinitely many polynomials c(x) over K such that a(x) + b(x) c(x) is irreducible of degree n, provided that K has a separable extension of degree n. Local view of a (complex) curve at a point of ramification 2 over the projective line. Ramification determines the Galois group of a cover. Conclusions and remarks: • The Main Theorem follows immediately from the combination of the results of section 3 and section 4 and since PAC fields are infinite. • The proof presented here is elementary and totally differs from proofs of the classical Dirichlet Theorem. • The polynomial c(x) in the main theorem is explicit up to a factor and can be computed using the Euclid Algorithm. • As infinite extensions of finite fields are PAC, our results holds for large finite fields. However, Kornblum (1919) already proved a Dirichlet’s Theorem for finite fields (analogously to the proof in the classical setup). • The theorems in this work are true in a more general setting, i.e., where the field K has a PAC extension M/K with a separable extension of degree n. • It seems that this method should produce irreducible polynomials of other type, e.g., c(x)2 - a where a has no square root in the field. Acknowledgments: My sincere thanks to Prof. Moshe Jarden, Prof. Peter Muller, Prof. Joseph Bernstein, and my adviser Prof. Dan Haran for illuminating discussions and numerous suggestions concerning the research. I would also like to thank my wife Hamutal for her help in creating this poster. The research reported here was partially carried out while I was a visitor at the Max-Planck-Institute fur Mathematik in Bonn.