An algebraic integer is a special kind of complex number. According to Wikipedia, in algebraic number theory, an algebraic integer is formally defined as follows:
An algebraic integer is a complex number that is a root of a monic polynomial (a polynomial whose leading coefficient is 1) with integer coefficients.
Understanding the Definition
Let's break down this definition:
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Complex Number: A number that can be expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit (√-1).
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Polynomial: An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents. For example, x² + 3x - 5 is a polynomial.
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Monic Polynomial: A polynomial in which the coefficient of the highest power of the variable is 1. For instance, x² + 3x - 5 is monic, but 2x² + 3x - 5 is not.
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Integer Coefficients: The coefficients of the polynomial are integers (..., -2, -1, 0, 1, 2, ...).
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Root of a Polynomial: A value that, when substituted for the variable in the polynomial, makes the polynomial equal to zero.
Examples
Here are some examples to illustrate the concept:
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Example 1: The Integer 2
The number 2 is an algebraic integer because it's a root of the monic polynomial x - 2 = 0.
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Example 2: √2
The square root of 2 (√2) is an algebraic integer because it's a root of the monic polynomial x² - 2 = 0.
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Example 3: i (the imaginary unit)
The imaginary unit i is an algebraic integer because it's a root of the monic polynomial x² + 1 = 0.
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Example 4: (1 + √5) / 2 (The Golden Ratio)
The golden ratio, (1 + √5) / 2, is an algebraic integer because it's a root of the monic polynomial x² - x - 1 = 0.
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Non-Example: 1/2
The rational number 1/2 is not an algebraic integer. While it's a root of the polynomial 2x - 1 = 0, this polynomial is not monic. There's no monic polynomial with integer coefficients that has 1/2 as a root.
Key Properties
- All integers are algebraic integers.
- The sum, difference, and product of two algebraic integers are also algebraic integers. This means the algebraic integers form a ring.
- A rational number is an algebraic integer if and only if it is an integer.
Why Are They Important?
Algebraic integers are fundamental in algebraic number theory because they generalize the concept of integers from the rational numbers to other number fields. They allow us to study the arithmetic properties of these fields in a similar way to how we study the arithmetic of the integers.
