Algebraic integer

In algebraic number theory, an algebraic integer is a special kind of complex number. It is a number that can be a root of a certain type of equation called a monic polynomial. This means the polynomial’s highest degree term has a coefficient of 1, and all other coefficients are whole numbers.

All algebraic integers form a group under addition, subtraction, and multiplication. This makes them very useful in mathematics. Every algebraic integer is part of something called the ring of integers of a number field. This helps mathematicians study number theory in a deeper way.

Algebraic integers are important because they help us understand the properties of numbers and their relationships. They connect many areas of math, showing how different number systems fit together.

Definitions

An algebraic integer is a special kind of number that comes from solving certain equations. Imagine you have a polynomial, which is a math expression like (x^2 - 3x + 2), where the highest power of (x) has a coefficient of 1 (this is called "monic"), and all the numbers filling in the polynomial are whole numbers (like 1, 2, -3, etc.). If a complex number helps solve such a polynomial (meaning when you plug it in, the whole thing equals zero), then that number is called an algebraic integer.

Algebraic integers behave nicely with basic math — you can add, subtract, or multiply them and still get another algebraic integer. They are important in higher math because they help us understand the structure of numbers better.

Examples

The only algebraic integers found in simple fractions are the whole numbers. A fraction like a/b is only an algebraic integer if b divides a evenly.

The square root of a whole number is an algebraic integer, but it is not a simple number unless the number is a perfect square.

If d is a whole number that cannot be divided by another whole number squared, then special number systems can be built using the square root of d. These systems include certain special numbers that are algebraic integers. For example, when d meets certain conditions, the number made from 1/2 times (1 + square root of d) is also an algebraic integer.

If α is an algebraic integer, then the nth root of α is also an algebraic integer. This means that special numbers built from α will still follow the same rules.

Finite generation of ring extension

For any number α, the way we can build new numbers from the regular whole numbers (like 1, 2, 3) by using α is called a ring extension. This special building process is called "finitely generated" only when α is an algebraic integer.

This idea connects closely to how we study algebraic numbers, but instead of using fractions, we use whole numbers. The main difference is that we only use positive powers of α when building these new numbers. Both algebraic integers and algebraic numbers are defined by being solutions to special equations with whole number or fraction coefficients.

Ring

The sum, difference, and product of two algebraic integers is always another algebraic integer. However, their quotient might not be an algebraic integer. This means that algebraic integers form a special kind of structure called a ring.

This idea can be shown using similar steps as for algebraic numbers, but replacing certain numbers with integers.

Additional facts

Any number made from whole numbers using roots, addition, and multiplication is an algebraic integer, but many algebraic integers are not like this. This idea is connected to the Abel–Ruffini theorem.

If the main part of the special math rule for an algebraic integer is 1 or -1, then flipping that number around still gives another algebraic integer.

If a special number comes from a math rule with whole number parts, then changing it in a certain way gives another algebraic integer.

Every special number can be written as a piece of an algebraic integer divided by another piece of an algebraic integer. In fact, we can always pick the bottom piece to be a whole number that is positive.

The only regular whole-number algebraic integers are the whole numbers themselves. This comes from a math rule for special equations with a main part of 1.