CHAPTER 1
Number Fields
1. Example : Quadratic number f i elds
Before we consider number f i elds in general, let us begin with the fairly concrete
case of quadratic number f i elds. A quadratic number f i eld is an extension K of Q
of degree 2. The fundamental examples (in fact, as we shall see in a moment the
only example) are f i elds of the form
Q( √ d) = {a + b √ d | a, b ∈ Q}
where d ∈ Q is not the square of another rational number.
There is an issue that arises as soon as we write down these f i elds, and it is
important that we deal with it immediately: what exactly do we mean by
√ d?
There are several possible answers to this question. The most obvious is that by
√ d we mean a specif i c choice of a complex square root of d. Q( √ d) is then def i ned
as a subf i eld of the complex numbers. The dif f i culty with this is that the notation
“ √ d” is ambiguous; d has two complex square roots, and there is no algebraic way
to tell them apart.
Algebraists have a standard way to avoid this sort of ambiguity; we can simply
def i ne
Q( √ d) = Q[x]/(x 2 - d).
There is no ambiguity with this notation;
√ d really means x, and x behaves as a
formal algebraic object with the property that x 2 = d.
This second def i nition is somehow the algebraically correct one, as there is no
ambiguity and it allows Q( √ d) to exist completely independently of the complex
numbers. However, it is far easier to think about Q( √ d) as a subf i eld of the complex
numbers. The ability to think of Q( √ d) as a subf i eld of the complex numbers also
becomes important when one wishes to compare f i elds Q( √ d 1 ) and Q( √ d 2 ) for
two dif f erent numbers d 1 and d 2 ; the abstract algebraic f i elds Q[x]/(x 2 - d 1 ) and
Q[y]/(y 2 -d 2 ) have no natural relation to each other, while these same f i elds viewed
as subf i elds of C can be compared more easily.
The best approach, then, seems to be to pretend to follow the formal algebraic
option, but to actually view everything as subf i elds of the complex numbers. We
can do this through the notion of a complex embedding; this is simply an injection
σ : Q[x]/(x 2 - d) → C.
As we have already observed, there are exactly two such maps, one for each complex
square root of d.
Before we continue we really ought to decide which complex number we mean
by
√ d. There is unfortunately no consistent way to do this, in the sense that we
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