Английская Википедия:Algebraic integer
Шаблон:Short description Шаблон:About Шаблон:Distinguish Шаблон:Use mdy dates Шаблон:Use American English In algebraic number theory, an algebraic integer is a complex number that is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers Шаблон:Mvar is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers.
The ring of integers of a number field Шаблон:Mvar, denoted by Шаблон:Math, is the intersection of Шаблон:Mvar and Шаблон:Mvar: it can also be characterised as the maximal order of the field Шаблон:Mvar. Each algebraic integer belongs to the ring of integers of some number field. A number Шаблон:Mvar is an algebraic integer if and only if the ring <math>\mathbb{Z}[\alpha]</math> is finitely generated as an abelian group, which is to say, as a <math>\mathbb{Z}</math>-module.
Definitions
The following are equivalent definitions of an algebraic integer. Let Шаблон:Mvar be a number field (i.e., a finite extension of <math>\mathbb{Q}</math>, the field of rational numbers), in other words, <math>K = \Q(\theta)</math> for some algebraic number <math>\theta \in \Complex</math> by the primitive element theorem.
- Шаблон:Math is an algebraic integer if there exists a monic polynomial <math>f(x) \in \Z[x]</math> such that Шаблон:Math.
- Шаблон:Math is an algebraic integer if the minimal monic polynomial of Шаблон:Mvar over <math>\mathbb{Q}</math> is in <math>\Z[x]</math>.
- Шаблон:Math is an algebraic integer if <math>\Z[\alpha]</math> is a finitely generated <math>\Z</math>-module.
- Шаблон:Math is an algebraic integer if there exists a non-zero finitely generated <math>\Z</math>-submodule <math>M \subset \Complex</math> such that Шаблон:Math.
Algebraic integers are a special case of integral elements of a ring extension. In particular, an algebraic integer is an integral element of a finite extension <math>K / \mathbb{Q}</math>.
Examples
- The only algebraic integers that are found in the set of rational numbers are the integers. In other words, the intersection of <math>\mathbb{Q}</math> and Шаблон:Mvar is exactly <math>\mathbb{Z}</math>. The rational number Шаблон:Math is not an algebraic integer unless Шаблон:Mvar divides Шаблон:Mvar. Note that the leading coefficient of the polynomial Шаблон:Math is the integer Шаблон:Mvar. As another special case, the square root <math>\sqrt{n}</math> of a nonnegative integer Шаблон:Mvar is an algebraic integer, but is irrational unless Шаблон:Mvar is a perfect square.
- If Шаблон:Mvar is a square-free integer then the extension <math>K = \mathbb{Q}(\sqrt{d}\,)</math> is a quadratic field of rational numbers. The ring of algebraic integers Шаблон:Math contains <math>\sqrt{d}</math> since this is a root of the monic polynomial Шаблон:Math. Moreover, if Шаблон:Math, then the element <math display=inline>\frac{1}{2}(1 + \sqrt{d}\,)</math> is also an algebraic integer. It satisfies the polynomial Шаблон:Math where the constant term Шаблон:Math is an integer. The full ring of integers is generated by <math>\sqrt{d}</math> or <math display=inline>\frac{1}{2}(1 + \sqrt{d}\,)</math> respectively. See Quadratic integer for more.
- The ring of integers of the field <math>F = \Q[\alpha]</math>, Шаблон:Math, has the following integral basis, writing Шаблон:Math for two square-free coprime integers Шаблон:Mvar and Шаблон:Mvar:[1] <math display="block">\begin{cases}
1, \alpha, \dfrac{\alpha^2 \pm k^2 \alpha + k^2}{3k} & m \equiv \pm 1 \bmod 9 \\ 1, \alpha, \dfrac{\alpha^2}k & \text{otherwise} \end{cases}</math>
- If Шаблон:Mvar is a primitive Шаблон:Mvarth root of unity, then the ring of integers of the cyclotomic field <math>\Q(\zeta_n)</math> is precisely <math>\Z[\zeta_n]</math>.
- If Шаблон:Mvar is an algebraic integer then Шаблон:Math is another algebraic integer. A polynomial for Шаблон:Mvar is obtained by substituting Шаблон:Math in the polynomial for Шаблон:Mvar.
Non-example
- If Шаблон:Math is a primitive polynomial that has integer coefficients but is not monic, and Шаблон:Mvar is irreducible over <math>\mathbb{Q}</math>, then none of the roots of Шаблон:Mvar are algebraic integers (but are algebraic numbers). Here primitive is used in the sense that the highest common factor of the coefficients of Шаблон:Mvar is 1; this is weaker than requiring the coefficients to be pairwise relatively prime.
Facts
- The sum, difference and product of two algebraic integers is an algebraic integer. In general their quotient is not. The monic polynomial involved is generally of higher degree than those of the original algebraic integers, and can be found by taking resultants and factoring. For example, if Шаблон:Math, Шаблон:Math and Шаблон:Math, then eliminating Шаблон:Mvar and Шаблон:Mvar from Шаблон:Math and the polynomials satisfied by Шаблон:Mvar and Шаблон:Mvar using the resultant gives Шаблон:Math, which is irreducible, and is the monic equation satisfied by the product. (To see that the Шаблон:Mvar is a root of the Шаблон:Mvar-resultant of Шаблон:Math and Шаблон:Math, one might use the fact that the resultant is contained in the ideal generated by its two input polynomials.)
- Any number constructible out of the integers with roots, addition, and multiplication is therefore an algebraic integer; but not all algebraic integers are so constructible: in a naïve sense, most roots of irreducible quintics are not. This is the Abel–Ruffini theorem.
- Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer. In other words, the algebraic integers form a ring that is integrally closed in any of its extensions.
- The ring of algebraic integers is a Bézout domain, as a consequence of the principal ideal theorem.
- If the monic polynomial associated with an algebraic integer has constant term 1 or −1, then the reciprocal of that algebraic integer is also an algebraic integer, and is a unit, an element of the group of units of the ring of algebraic integers.
- Every algebraic number can be written as the ratio of an algebraic integer to a non-zero algebraic integer. In fact, the denominator can always be chosen to be a positive integer. Specifically, if Шаблон:Math is an algebraic number that is a root of the polynomial Шаблон:Math with integer coefficients and leading term Шаблон:Math for Шаблон:Math then Шаблон:Math is the promised ratio. In particular, Шаблон:Math is an algebraic integer because it is a root of Шаблон:Math, which is a monic polynomial in Шаблон:Math with integer coefficients.
See also
- Integral element
- Gaussian integer
- Eisenstein integer
- Root of unity
- Dirichlet's unit theorem
- Fundamental units
References
Шаблон:Refend Шаблон:Algebraic numbers
