Chapter 4. Algebraic Number Theory
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...
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.