Algebraic integer

Template:Short description Template:About Template:Distinguish Template:Use mdy dates Template: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 Template: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 Template:Mvar, denoted by Template:Math, is the intersection of Template:Mvar and Template:Mvar: it can also be characterised as the maximal order of the field Template:Mvar. Each algebraic integer belongs to the ring of integers of some number field. A number Template: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 Template: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.

• Template:Math is an algebraic integer if there exists a monic polynomial <math>f(x) \in \Z[x]</math> such that Template:Math.
• Template:Math is an algebraic integer if the minimal monic polynomial of Template:Mvar over <math>\mathbb{Q}</math> is in <math>\Z[x]</math>.
• Template:Math is an algebraic integer if <math>\Z[\alpha]</math> is a finitely generated <math>\Z</math>-module.
• Template:Math is an algebraic integer if there exists a non-zero finitely generated <math>\Z</math>-submodule <math>M \subset \Complex</math> such that Template: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 Template:Mvar is exactly <math>\mathbb{Z}</math>. The rational number Template:Math is not an algebraic integer unless Template:Mvar divides Template:Mvar. The leading coefficient of the polynomial Template:Math is the integer Template:Mvar.
• The square root <math>\sqrt{n}</math> of a nonnegative integer Template:Mvar is an algebraic integer, but is irrational unless Template:Mvar is a perfect square.
• If Template: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 Template:Math contains <math>\sqrt{d}</math> since this is a root of the monic polynomial Template:Math. Moreover, if Template:Math, then the element <math display=inline>\frac{1}{2}(1 + \sqrt{d}\,)</math> is also an algebraic integer. It satisfies the polynomial Template:Math where the constant term Template: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>, Template:Math, has the following integral basis, writing Template:Math for two square-free coprime integers Template:Mvar and Template:Mvar:<ref>Template:Cite book</ref> <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 Template:Mvar is a primitive Template: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 Template:Mvar is an algebraic integer then Template:Math is another algebraic integer. A polynomial for Template:Mvar is obtained by substituting Template:Math in the polynomial for Template:Mvar.

Non-example

• If Template:Math is a primitive polynomial that has integer coefficients but is not monic, and Template:Mvar is irreducible over <math>\mathbb{Q}</math>, then none of the roots of Template:Mvar are algebraic integers (but are algebraic numbers). Here primitive is used in the sense that the highest common factor of the coefficients of Template:Mvar is 1, which is weaker than requiring the coefficients to be pairwise relatively prime.

Finite generation of ring extension

For any Template:Math, the ring extension (in the sense that is equivalent to field extension) of the integers by Template:Math, denoted by <math>\Z[\alpha] \equiv \left\{\sum_{i=0}^n \alpha^i z_i | z_i\in \Z, n\in \Z\right\}</math>, is finitely generated if and only if Template:Math is an algebraic integer.

The proof is analogous to that of the corresponding fact regarding algebraic numbers, with <math>\Q</math> there replaced by <math>\Z</math> here, and the notion of field extension degree replaced by finite generation (using the fact that <math>\Z</math> is finitely generated itself); the only required change is that only non-negative powers of Template:Math are involved in the proof.

The analogy is possible because both algebraic integers and algebraic numbers are defined as roots of monic polynomials over either <math>\Z</math> or <math>\Q</math>, respectively.

Ring

The sum, difference and product of two algebraic integers is an algebraic integer. In general their quotient is not. Thus the algebraic integers form a ring.

This can be shown analogously to the corresponding proof for algebraic numbers, using the integers <math>\Z</math> instead of the rationals <math>\Q</math>.

One may also construct explicitly the monic polynomial involved, which is generally of higher degree than those of the original algebraic integers, by taking resultants and factoring. For example, if Template:Math, Template:Math and Template:Math, then eliminating Template:Mvar and Template:Mvar from Template:Math and the polynomials satisfied by Template:Mvar and Template:Mvar using the resultant gives Template:Math, which is irreducible, and is the monic equation satisfied by the product. (To see that the Template:Mvar is a root of the Template:Mvar-resultant of Template:Math and Template:Math, one might use the fact that the resultant is contained in the ideal generated by its two input polynomials.)

Integral closure

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.

Again, the proof is analogous to the corresponding proof for algebraic numbers being algebraically closed.

Additional facts

• Any number constructible out of the integers with roots, addition, and multiplication is 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.
• 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 each is a unit, an element of the group of units of the ring of algebraic integers.
• If Template:Math is an algebraic number then Template:Math is an algebraic integer, where Template:Mvar satisfies a polynomial Template:Math with integer coefficients and where Template:Math is the highest-degree term of Template:Math. The value Template:Math is an algebraic integer because it is a root of Template:Math, where Template:Math is a monic polynomial with integer coefficients.
• If Template:Math is an algebraic number then it 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. The ratio is Template:Math, where Template:Mvar satisfies a polynomial Template:Math with integer coefficients and where Template:Math is the highest-degree term of Template:Math.
• The only rational algebraic integers are the integers. That is, if Template:Mvar is an algebraic integer and <math>x\in\Q</math> then <math>x\in\Z</math>. This is a direct result of the rational root theorem for the case of a monic polynomial.

See also

• Gaussian integer
• Eisenstein integer
• Root of unity
• Dirichlet's unit theorem
• Fundamental units

References

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• Template:Cite book