Eisenstein integer

The Eisenstein numbers are a generalization of the integers to the complex numbers. They are after the German mathematician Gotthold Eisenstein, a student of Gauss named. The Gaussian numbers are another generalization of the integers to the complex numbers. The Eisenstein numbers are the whole ring, ie, the maximal order of the quadratic number field which coincides with the cyclotomic field. They occur for example in the formulation of the cubic reciprocity law (→ see cubic reciprocity law in this article).

Definition

A complex number is an Eisenstein - number if they are in the form

And integers and represent leaves. is a (primitive ) third root of unity, and thus satisfies the equation

In other words, the Eisenstein numbers form the ring, which arises from the ring of integers by adjoining the primitive third root of unity. The whole ring of the cyclotomic field that arises from by adjoining a primitive sixth root of unity, for example, by adjoining the main value, is also consistent with the Eisenstein numbers.

Geometrical Meaning

The Eisenstein numbers form a triangular lattice in the Gaussian number plane. They correspond to the centers of densest sphere packing in two dimensions.

Number Theory

On the Eisenstein numbers to number theory can operate: the units are exactly the six complex roots of the equation, the cyclic unit group is thus generated by each primitive sixth root of unity, for example. At any Eisenstein integers, which is different from 0, exist exactly 6 associated elements that form the coset.

One can define prime elements analogous to the primes in and show that the prime factorization of a number Eisenstein - except for Associated awareness and order of the prime factors - is unique, that the Eisenstein numbers form a factorial integral domain. Integers of the form are always separable in the Eisenstein numbers. Therefore, the numbers 3, 7, 13, 19, ... not primes in the figures Eisenstein.

More specifically, the following three cases occur:

  • 3 is a special case. This is the only prime number in that by the square of a Primelementes in divisible. It is said in algebraic number theory, this prime number is branched.
  • Positive primes that satisfy the congruence are also primes. We then say, these primes are sluggish.
  • Positive primes that satisfy the congruence to be products of two mutually complex conjugate prime elements in then said these primes are disassembled.

The contracts are prime numbers, then, and a prime factorization of the first split primes is:

The 6 associated elements of a Primelementes are prime, as is the prime element to a complex conjugate element.

Since the norm of an element of always is in that form, the sluggish all primes and the prime elements that occur as factors in the decomposition of the disassembled all prime numbers, along with their Associate the set of all prime elements in.

The ring of Eisenstein numbers is Euclidean.

Cubic residual character

In the ring of Eisenstein numbers one sentence which is analogous to Fermat's little theorem of elementary number theory applies:

If now is true for the standard of that, and so is, then a power exponent integer- and it is

These are called unit root a cubic residue character of modulo and writes for

The designation as a character arises from the fact that the mapping for a fixed prime element defines a unitary character of the multiplicative group of the body.

The congruence is solvable if and only if. If the congruence solvable and, then is called a cubic residue modulo the congruence is permanent, a cubic non-residue modulo. Similarly, the terms cubic residue and non-residue are generally declared when no prime element to but is prime.

The cubic residual character of primes, which can not be associated, formal properties similar to the properties of the Legendre symbol:

The cubic residue character may continue in the " denominator " multiplicative effect on composite numbers that are relatively prime to 3. It is then defined in addition, that the so- defined cubic residue symbol has the value 0 if the numbers are not relatively prime in the ring of Eisenstein numbers, but is relatively prime to 3. This generalization is analogous to the generalization of the Legendre symbol to the Jacobi symbol, except for the fact that that the following applies for the case or equivalent to that of the standard is divided into 3, no value for the symbol is defined. - Sometimes in the latter case, the symbol 0 is set. This variant does not alter the following statements.

Similar to the Jacobi symbol are for a " denominator " of the cubic residue symbol which is not a prime element, the following statements:

  • By the multiplicative continued by the definition:
  • Is the " counter" a cubic residue modulo and then the symbol takes the value 1.
  • If the symbol to a number other than 1, then the counter is not a cubic residue modulo or not relatively prime to 3
  • The icon may take a value of 1, even if the counter is a cubic non-residue modulo.

Primary numbers

For the formulation of a cubic reciprocity law on the ring of Eisenstein numbers certain representatives must be selected from the Associates of Eisenstein integers. Eisenstein calls a number primary if it satisfies the congruence. One can easily prove that an associate element to them is accurate for numbers whose norm ( in ) prime to 3, primarily for the purposes of this definition. A disadvantage of the definition is that the product of two numbers of primary is always the subject number of a primary figure.

We define therefore mostly today:

  • An iron stone number is primarily if it is relatively prime to 3 and is congruent modulo a common integer.

This definition is equivalent to the fact that the congruence is valid in the ring of numbers Eisenstein. We then have:

Since -1 is always a cubic residue, the uniqueness of this definition goes " up to sign " for the formulation of the law of reciprocity from.

Cubic reciprocity law

For two primary numbers

For this there are cubic reciprocity law sets to extend the units and the prime element:

If is primary and is then also applies

For primary " denominator " with can be replaced by the associated primary element, without changing the value of the symbol.

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