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

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

For example,

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

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

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

For example,

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

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

Division is also possible because

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 $K$:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

The Tower Law

Field extensions may be stacked. If

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

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

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,

Here

and

Therefore

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

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

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

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

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.