Fibonacci Fields

In this paper, we consider fields determined by the /1 roots of the zeros a and fi of the polynomial x x 1 ; a is the positive zero. The tools for studying these fields will include the Fibonacci and Lucas polynomials. Generalized versions of Fibonacci and Lucas polynomials have been studied in [1], [2], [3], [4], [5], [6], [7], and [12], among others. For the most part, these generalizations consist of considering roots of more general quadratic equations that also satisfy Binet identities. However, it is just the simplest version of these polynomials that we shall need for the results in this paper. (For a far-reaching generalization of all of these generalizations in the context of multiplicative arithmetic functions, see [9].) These polynomials determine many of the properties of the root fields; e.g., they provide the defining polynomials for those fields; they yield a collection of algebraic integers which behave like the Fibonacci numbers and the Lucas numbers in the ring of rational integers; they determine the discriminants of these fields; and, they provide a means of embedding which gives the lattice structure of the fields. In Part 1, we list properties of these polynomials which we shall need later. In Part 2, the (odd) m® roots of a and ft are discussed; the constant am which is, essentially, the sum of two conjugate roots, is introduced. One of two important theorems here is Theorem 2.1, which tells us that the m^ Lucas polynomial evaluated at am is, up to sign, equal to 1. This will enable us to define a new set of polynomials (by adding a constant to the Lucas polynomial) which, in Part 4, will turn out to be irreducible over the rationals and, hence, will provide us with some useful extension fields (Theorem 4.2). The other important theorem in Part 2 is Theorem 2.2, which tells us that the w* Lucas polynomial evaluated at amn is an. This theorem will lead to an embedding theorem for our fields in Part 4 (Lemma 4.2.2). In Part 3, we introduce numbers in our extension fields generalizing the Fibonacci numbers, which are algebraic integers in these fields and which turn out to have a peculiar quasi-periodic behavior (Theorem 3.4). (In a sequel to [9], this behavior will be seen to be one typically associated with arithmetic functions.) In Part 4, the lattice structure of this family of fields is investigated (Lemma 4.2.2, Corollary 4.2.3, Theorem 4.3). Theorem 4.4 tells us that it is the Fibonacci polynomials which provide us with the discriminants of our fields. The remainder of the paper is occupied with some calculations using a well-known matrix representation of the fields, illustrating computations which produce units and primes in these fields. The author is indebted to the referee for many helpful suggestions for which he is grateful; especially, he would like to thank the referee for calling to his' attention the rich theory of quadratic fields of' Richaud-Degert type and of R. A. Mollin's book [10]. The fields studied here are extensions of a field of this type.