Algebraic number field facts for kids

Kids Encyclopedia Facts

An algebraic number field is a special kind of field in mathematics. Think of it as a collection of numbers where you can add, subtract, multiply, and divide, just like with regular rational numbers (fractions). What makes it special is that it's built by adding "algebraic numbers" to the rational numbers.

Imagine you start with all the fractions, like 1/2, -3, 7/4. This is the field of rational numbers, often called Q. An algebraic number field, let's call it K, is a bigger field that contains all of Q plus some other numbers that are "algebraic."

What's an algebraic number? It's any number that can be the solution to a polynomial equation with integer coefficients. For example, the square root of 2 (√2) is an algebraic number because it's a solution to the equation x² - 2 = 0. You can't write √2 as a simple fraction.

Algebraic number fields are important in a branch of math called algebraic number theory. This field helps us understand the hidden structures behind numbers.

What is an Algebraic Number Field?

To understand algebraic number fields, we need to know about two basic ideas: fields and vector spaces.

Understanding Fields

A field is a set of numbers where you can do addition, subtraction, multiplication, and division (except by zero). These operations must follow certain rules, like the order of operations. The most common example is the set of rational numbers (fractions), called Q.

What are Vector Spaces?

You can think of a vector space as a collection of "vectors." A vector can be like a list of numbers, for example, (x, y, z). You can add these lists together, and you can multiply them by a single number (called a "scalar") from a field.

For example, if you have vectors (x1, x2) where x1 and x2 are rational numbers, you can add them: (x1, x2) + (y1, y2) = (x1+y1, x2+y2). You can also multiply by a rational number 'c': c * (x1, x2) = (c*x1, c*x2).

A vector space has a "dimension." This is like saying how many numbers you need in your list to describe any vector in that space. If the list is always a fixed, finite length (like (x1, x2, x3)), then it's a "finite-dimensional" vector space.

Defining an Algebraic Number Field

An algebraic number field (or just number field) is a field K that contains the rational numbers Q. When you think of K as a vector space over Q, it must have a finite dimension.

This "finite dimension" is also called the "degree" of the field extension. It means that every number in K can be written as a combination of a limited set of basic numbers from K, using only rational numbers for multiplication.

Examples of Number Fields

Rational Numbers (Q): This is the simplest number field. It's like a number field of dimension 1 over itself. Many properties of other number fields are compared to Q.
Gaussian Rationals (Q(i)): This field includes numbers like a + bi, where a and b are rational numbers, and i is the imaginary unit (where i² = -1). You can add, subtract, and multiply these numbers. For example:

* (1 + 2i) + (3 + 4i) = (1+3) + (2+4)i = 4 + 6i * (1 + 2i) * (3 + 4i) = 1*3 + 1*4i + 2i*3 + 2i*4i = 3 + 4i + 6i + 8i² = 3 + 10i - 8 = -5 + 10i Every non-zero Gaussian rational number can be divided. This field has a dimension of 2 when seen as a vector space over Q.

Quadratic Fields (Q(√d)): These are fields like Q(√2) or Q(√-5). They are formed by adding the square root of a square-free integer d to the rational numbers. Numbers in these fields look like a + b√d, where a and b are rational numbers.
Cyclotomic Fields (Q(ζn)): These fields are made by adding special numbers called "roots of unity" to Q. A root of unity is a number that, when raised to a certain power n, equals 1. For example, if n=4, then i is a 4th root of unity (i⁴ = 1).

What are NOT Number Fields?

Real Numbers (R) and Complex Numbers (C): These fields are much larger than any number field. They have an "infinite dimension" as vector spaces over Q. This means you can't describe all real or complex numbers using a finite list of basic numbers from Q.
Q² (Ordered Pairs of Rationals): This is the set of pairs (x, y) where x and y are rational numbers. While you can add and multiply them, it's not a field because you can multiply two non-zero pairs and get zero. For example, (1, 0) * (0, 1) = (0, 0). In a true field, this can't happen unless one of the original numbers was zero.

Algebraic Integers

In an algebraic number field K, every number is an algebraic number. This means it's a solution to a polynomial equation with rational numbers as coefficients. For example, if x is in K, then x is a root of an equation like: e_m x^m + e_{m-1} x^{m-1} + ... + e_1 x + e_0 = 0 where the e values are rational numbers.

If we can make the first coefficient (e_m) equal to 1, and all other coefficients are integers (whole numbers), then x is called an algebraic integer.

Any regular integer (like 5 or -10) is an algebraic integer. It's a solution to x - 5 = 0.
The sum and product of any two algebraic integers are also algebraic integers.
The collection of all algebraic integers within a number field K forms a special structure called a ring. This is known as the ring of integers of K, written as O_K. It's like the integer version of the number field.

Unique Factorization

In the world of regular integers (Z), every number can be broken down into a unique set of prime numbers (like 6 = 2 * 3). This is called unique factorization.

However, in the ring of integers O_K for some number fields, this unique factorization doesn't always happen for numbers themselves. For example, in the field Q(√-5), the number 6 can be factored in two different ways:

6 = 2 * 3
6 = (1 + √-5) * (1 - √-5)

These two factorizations are truly different; you can't just rearrange the parts or multiply by a simple "unit" (like -1 or 1) to get from one to the other.

This "failure" of unique factorization is an important topic in algebraic number theory. It's measured by something called the class number. If the class number is 1, then unique factorization holds for numbers in that field's ring of integers.

Bases for Number Fields

Just like a vector space has a basis (a set of vectors that can be used to build all other vectors), a number field also has bases.

Integral Basis

For a number field K of degree n, an integral basis is a set of n algebraic integers {b1, b2, ..., bn} from K. Any algebraic integer in O_K can be written as a unique combination of these basis elements, using only regular integers as multipliers. For example: x = m1*b1 + m2*b2 + ... + mn*bn, where m1, m2, ... are integers.

Power Basis

A power basis is a special kind of basis that looks like {1, x, x², ..., x^(n-1)} for some element x in K. Not all number fields have a power basis where x is an algebraic integer and the basis works for the entire ring of integers.

Trace and Norm

For any number x in a number field K, we can associate it with a special matrix. This matrix helps us understand how multiplying by x affects other numbers in the field.

From this matrix, we can calculate two important values:

The trace of x (written as Tr(x)): This is the sum of the numbers on the main diagonal of the matrix. It's like a special sum related to x.
The norm of x (written as N(x)): This is the determinant of the matrix. It's like a special product related to x.

These values (trace and norm) are always rational numbers. They are useful tools for studying the properties of numbers within the field.

The discriminant of a number field is another important value. It's an integer calculated from the trace of the basis elements. It's a unique property of the field itself, no matter which integral basis you choose.

Places

In mathematics, a "place" of a number field is a way to measure the "size" of numbers in that field. It's like a generalized absolute value.

For the rational numbers Q, there are two main types of places:

The usual absolute value (like |3| = 3, |-5| = 5). This leads to the real numbers R when you "complete" Q using this measure.
The p-adic absolute value (for any prime number p). This is a bit strange: numbers that are divisible by p become "smaller." For example, using the 5-adic absolute value, 5 is smaller than 1, 25 is even smaller, and so on. This leads to the p-adic numbers Qp when completed.

For a general number field K, there are also two types of places:

Archimedean places: These are like the usual absolute value and lead to real or complex numbers when you complete the field.
Non-Archimedean (or ultrametric) places: These are like the p-adic absolute values. Each non-Archimedean place corresponds to a unique prime ideal in the ring of integers O_K.

Ramification

Ramification is a concept that describes how prime numbers (or more generally, prime ideals) behave when you move from a smaller number field to a larger one.

Imagine you have a prime number p in the rational numbers Q. When you extend to a larger number field K, this prime p can "split" into several prime ideals in the ring of integers O_K.

For example, the prime number 2 in Q splits in the Gaussian rationals Q(i) as 2 = (1+i)(1-i).

Sometimes, when a prime splits, one or more of the new prime ideals might have a "power" greater than 1 in the factorization. This is called ramification. If a prime p ramifies, it means that the way it splits in the larger field is "special" or "less simple" than usual.

The discriminant of a number field is very important here. A prime number p ramifies in a number field K if and only if p divides the discriminant of K. This is a powerful theorem called the Dedekind discriminant theorem. It tells us exactly which primes will behave in this special, ramified way.

Galois Groups

The Galois group of a field extension helps us understand the symmetries of the numbers in the field. It's a group of special functions (called automorphisms) that rearrange the numbers in the field but keep the original smaller field fixed.

For example, the Galois group of Q(√2) over Q has two functions: one that leaves √2 as √2, and one that changes √2 to -√2. This group helps us understand how the field is structured.

Galois theory is a deep and important part of mathematics that connects field extensions with groups, allowing mathematicians to use group theory to solve problems about numbers.

Local-Global Principle

The "local-global principle" is a big idea in number theory. It suggests that if you want to solve a problem about numbers in a number field (a "global" problem), you can sometimes solve it by looking at the problem in all the "local" versions of the field.

The "local" versions are the completions of the number field at all its places (both Archimedean and non-Archimedean). These local fields are often simpler to work with because they have a more structured environment.

The Hasse principle is a famous example of this. It says that for certain types of equations (like quadratic equations), if the equation has a solution in all the local completions of a number field, then it must also have a solution in the number field itself. This isn't true for all equations, but it's a powerful tool when it applies.

This idea of moving from local information to global understanding is a central theme in modern number theory.