# Chapter 4. Algebraic Number Theory

## Fields Inside Larger Fields

A field is a number system in which addition, subtraction, multiplication, and division by nonzero elements are always possible. The rational numbers $\mathbb{Q}$, the real numbers $\mathbb{R}$, the complex numbers $\mathbb{C}$, and the finite fields $\mathbb{F}_p$ are basic examples.

A field extension occurs when one field is contained inside another.

**Definition.** Let $K$ and $L$ be fields. We say that $L$ is a field extension of $K$ if $K\subseteq L$ and the operations of $K$ agree with the operations inherited from $L$. In this case we write

$$
L/K
$$

and call $K$ the base field and $L$ the extension field.

For example,

$$
\mathbb{Q}\subseteq \mathbb{R}\subseteq \mathbb{C}.
$$

Thus $\mathbb{R}/\mathbb{Q}$, $\mathbb{C}/\mathbb{R}$, and $\mathbb{C}/\mathbb{Q}$ are field extensions.

Field extensions allow us to enlarge a field so that equations which previously had no solutions acquire solutions. The equation

$$
x^2-2=0
$$

has no solution in $\mathbb{Q}$, but it has solutions in $\mathbb{R}$, namely $\sqrt{2}$ and $-\sqrt{2}$. Thus adjoining $\sqrt{2}$ to $\mathbb{Q}$ gives a larger field in which the equation can be solved.

## Adjoining Elements

Given a field $K$ and an element $\alpha$ lying in some larger field, the notation

$$
K(\alpha)
$$

means the smallest field containing both $K$ and $\alpha$.

For example,

$$
\mathbb{Q}(\sqrt{2})
$$

is the smallest field containing the rational numbers and $\sqrt{2}$. Its elements are precisely the numbers

$$
a+b\sqrt{2},
\qquad a,b\in\mathbb{Q}.
$$

This set is closed under addition, subtraction, multiplication, and division by nonzero elements. For instance,

$$
(a+b\sqrt{2})(c+d\sqrt{2}) =
(ac+2bd)+(ad+bc)\sqrt{2}.
$$

Division is also possible because

$$
\frac{1}{a+b\sqrt{2}} =
\frac{a-b\sqrt{2}}{a^2-2b^2},
$$

provided $a+b\sqrt{2}\neq 0$. Since $a^2-2b^2\neq 0$ for rational $a,b$ unless $a=b=0$, the denominator is nonzero.

More generally, $K(\alpha)$ consists of all rational expressions in $\alpha$ with coefficients in $K$:

$$
K(\alpha) =
\left\{
\frac{f(\alpha)}{g(\alpha)}
:
f,g\in K[x],\ g(\alpha)\neq 0
\right\}.
$$

This description is useful because it connects field extensions with polynomials.

## Algebraic and Transcendental Elements

The behavior of $K(\alpha)$ depends strongly on whether $\alpha$ satisfies a polynomial equation over $K$.

**Definition.** Let $L/K$ be a field extension and let $\alpha\in L$. The element $\alpha$ is called algebraic over $K$ if there exists a nonzero polynomial $f(x)\in K[x]$ such that

$$
f(\alpha)=0.
$$

If no such polynomial exists, then $\alpha$ is called transcendental over $K$.

For example, $\sqrt{2}$ is algebraic over $\mathbb{Q}$, since it satisfies

$$
x^2-2=0.
$$

The complex number $i$ is algebraic over $\mathbb{R}$, since it satisfies

$$
x^2+1=0.
$$

The numbers $\pi$ and $e$ are transcendental over $\mathbb{Q}$, although proving this is highly nontrivial.

If $\alpha$ is algebraic over $K$, then among all nonzero polynomials in $K[x]$ that vanish at $\alpha$, there is a unique monic polynomial of smallest degree. This polynomial is called the minimal polynomial of $\alpha$ over $K$.

For $\sqrt{2}$ over $\mathbb{Q}$, the minimal polynomial is

$$
x^2-2.
$$

For $i$ over $\mathbb{R}$, the minimal polynomial is

$$
x^2+1.
$$

The minimal polynomial measures how complicated $\alpha$ is from the point of view of the base field.

## Degree of an Extension

Every field extension $L/K$ has a second structure: $L$ is a vector space over $K$. Addition is the addition in $L$, and scalar multiplication by elements of $K$ is the multiplication inherited from $L$.

**Definition.** The degree of a field extension $L/K$, denoted

$$
[L:K],
$$

is the dimension of $L$ as a vector space over $K$:

$$
[L:K]=\dim_K L.
$$

If this dimension is finite, then $L/K$ is called a finite extension.

For example, $\mathbb{Q}(\sqrt{2})$ has basis

$$
1,\sqrt{2}
$$

over $\mathbb{Q}$. Every element has the form $a+b\sqrt{2}$, and the representation is unique. Therefore

$$
[\mathbb{Q}(\sqrt{2}):\mathbb{Q}]=2.
$$

Similarly, $\mathbb{C}$ has basis $1,i$ over $\mathbb{R}$, so

$$
[\mathbb{C}:\mathbb{R}]=2.
$$

If $\alpha$ is algebraic over $K$ with minimal polynomial of degree $n$, then

$$
[K(\alpha):K]=n.
$$

Indeed, if the minimal polynomial has degree $n$, then every power of $\alpha$ of degree at least $n$ can be reduced to a $K$-linear combination of

$$
1,\alpha,\alpha^2,\ldots,\alpha^{n-1}.
$$

These $n$ elements form a basis of $K(\alpha)$ over $K$.

## The Tower Law

Field extensions may be stacked. If

$$
K\subseteq L\subseteq M,
$$

then $M/K$ is built by first extending $K$ to $L$, and then extending $L$ to $M$.

The degrees multiply.

**Theorem.** If $K\subseteq L\subseteq M$ and the degrees are finite, then

$$
[M:K]=[M:L][L:K].
$$

This is called the tower law.

To see the idea, suppose $u_1,\ldots,u_m$ is a basis of $M$ over $L$, and $v_1,\ldots,v_n$ is a basis of $L$ over $K$. Then the products

$$
u_i v_j
$$

form a basis of $M$ over $K$. There are $mn$ such products, so the dimension over $K$ is the product of the two intermediate dimensions.

For example,

$$
\mathbb{Q}\subseteq \mathbb{Q}(\sqrt{2})\subseteq \mathbb{Q}(\sqrt{2},i).
$$

Here

$$
[\mathbb{Q}(\sqrt{2}):\mathbb{Q}]=2
$$

and

$$
[\mathbb{Q}(\sqrt{2},i):\mathbb{Q}(\sqrt{2})]=2.
$$

Therefore

$$
[\mathbb{Q}(\sqrt{2},i):\mathbb{Q}]=4.
$$

## Field Extensions in Number Theory

Field extensions are central to modern number theory because arithmetic often becomes clearer after passing from $\mathbb{Q}$ to a larger field.

The equation

$$
x^2+y^2=z^2
$$

can be studied over the rational numbers, but equations involving expressions such as

$$
x^2+1
$$

naturally lead to the field $\mathbb{Q}(i)$. Similarly, the study of roots of unity leads to cyclotomic fields

$$
\mathbb{Q}(\zeta_n),
$$

where $\zeta_n$ is a primitive $n$-th root of unity.

Algebraic number theory studies finite extensions of $\mathbb{Q}$. These fields are called number fields. Their arithmetic generalizes the arithmetic of the rational integers. Instead of studying only

$$
\mathbb{Z}\subseteq \mathbb{Q},
$$

one studies the ring of algebraic integers inside a number field. This shift leads to ideals, class groups, units, ramification, and reciprocity laws.

Thus a field extension is not merely a larger field. It is a controlled enlargement of arithmetic, designed to make hidden structure visible.