Some Properties of Extensions of Small Degree over Q

This paper demonstrates a few characteristics of finite extensions of small degree over the rational numbers Q. It comprises attempts at classification of all such extensions of degrees 2, 3, and 4, as well as mention of how one might begin the process of classifying all finite fields with degree a power of 2. Some topics discussed are discriminants of certain fields, how to represent a given extension as a simple extension, and some applied Galois theory. 1. Quadratic Extensions It is fairly clear why quadratic extensions are in most respects the easiest to understand: they all have the same Galois group (Z2), their degree over Q is as small as possible without being trivial, and the number 2 is in many ways very easy to work with. 1.1. The Make-up of a Quadratic Number Ring. We begin by completely classifying the elements of the number ring associated with any number field of the form Q( √ m), where m 6= 1 or 0, is squarefree (i.e. a quadratic number field). This ring is by definition simply A ∩ Q(√m), where A is the set of algebraic integers, defined here to be the set {α ∈ C : ∃ f(x) ∈ Z[x]  f(α) = 0, f(x) monic and irreducible}. 1.1.1. Preliminaries. We know that [Q( √ m) : Q] = 2, and so any element of A ∩ Q( √ m) is going to have a minimum polynomial f(x) with deg f(x) ≤ 2. Thus we know that the algebraic numbers in Q( √ m) will simply be those elements α ∈ C such that f(α) = 0 for some f(x) ∈ Z, monic and irreducible, with deg f(x) ≤ 2. Our goal in section ?? will be to show that any root of such a polynomial can be expressed in one of the following ways: • as a + b√m, where a, b ∈ Z and m ≡ 2 or 3 (mod 4), for some squarefree m • or as a+b √ m 2 , where a, b ∈ Z, a ≡ b (mod 4), and m ≡ 1 (mod 4), for some squarefree m.