Eisenstein number

Eisenstein numbers as points of a triangular grid in the complex number plane

The Eisenstein numbers are a generalization of the whole numbers to the complex numbers . They are named after the German mathematician Gotthold Eisenstein , a student of Gauss . The Gaussian numbers are another generalization of the whole numbers to the complex numbers. The Eisenstein numbers are the whole ring , i.e. the maximum order of the square number field , which corresponds to the 3rd  circle division field . They occur, for example, in the formulation of the cubic reciprocity law (→ see cubic reciprocity law in this article). ${\ displaystyle \ mathbb {Q} \ left ({\ sqrt {-3}} \ right)}$ ${\ displaystyle \ mathbb {Q} (\ mu _ {3})}$

definition

A complex number is an Eisenstein number if it is in the shape ${\ displaystyle E}$

${\ displaystyle E = a + b \, \ omega}$ With ${\ displaystyle \ omega = e ^ {2 \ pi \ mathrm {i} / 3} = - {\ frac {1} {2}} + {\ frac {\ mathrm {i}} {2}} {\ sqrt {3}}}$

and whole numbers and can be represented. is a (primitive) third root of unity and thus satisfies the equation ${\ displaystyle a}$${\ displaystyle b}$${\ displaystyle \ omega}$

${\ displaystyle \ omega ^ {2} + \ omega + 1 = 0.}$

In the following, the primitive root of unity mentioned above always designates exactly , not the complex conjugate second zero of this quadratic equation. ${\ displaystyle \ omega}$${\ displaystyle \ omega}$

In other words: The Eisenstein numbers form the ring that arises from the ring of whole numbers through the adjunction of the primitive 3rd root of unity . The totality ring of the body of the circle, which arises from the adjunction of a primitive 6th root of unity , for example through the adjunction of the main value , also agrees with the Eisenstein numbers. ${\ displaystyle \ mathbb {Z} [\ omega]}$${\ displaystyle \ omega}$${\ displaystyle \ mathbb {Q}}$ ${\ displaystyle - \ omega ^ {2} = _ {H} {\ sqrt [{3}] {- 1}} = e ^ {\ pi \ mathrm {i} / 3}}$${\ displaystyle \ mathbb {Z} [- \ omega ^ {2}]}$

Geometric meaning

"Small" prime elements among the Eisenstein numbers in the complex number plane. The 60 ° rotational symmetry follows from the existence of six units in .${\ displaystyle \ mathbb {Z} [\ omega]}$

The Eisenstein numbers form a triangular grid in the Gaussian plane of numbers . They correspond to the centers of a closest packing of spheres in two dimensions.

Number theory

Number theory can be practiced on the Eisenstein numbers : The units are exactly the six complex zeros of the equation , the cyclic unit group is thus generated by each of the two primitive 6th roots of unit or . For each of the different Eisenstein numbers there are exactly six associated elements , which form a secondary class in the multiplicative group of the body . ${\ displaystyle X ^ {6} = 1}$${\ displaystyle U}$${\ displaystyle e ^ {+ 2 \ pi i / 6} = - \ omega ^ {2}}$${\ displaystyle e ^ {- 2 \ pi i / 6} = - \ omega}$${\ displaystyle 0}$${\ displaystyle \ alpha}$${\ displaystyle \ mathbb {Q} \ left ({\ sqrt {-3}} \ right)}$ ${\ displaystyle \ alpha U}$

One can define prime elements analogously to the prime numbers in and show that the prime factorization of an Eisenstein number is unambiguous - apart from the association and sequence of the prime factors. The Eisenstein numbers thus form a factorial integrity range . All whole numbers of the form can be decomposed into the Eisenstein numbers. So there the numbers 3, 7, 13, 19, ... are not prime elements. ${\ displaystyle \ mathbb {Z}}$${\ displaystyle m ^ {2} + 3n ^ {2}}$

The following three cases occur more precisely:

• 3 is a special case . This is the only prime number in that is divisible by the square of a prime element in . In algebraic number theory we say that this prime number is branched .${\ displaystyle 3 = - \ omega ^ {2} (1- \ omega) ^ {2}}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle \ mathbb {Z} [\ omega]}$
• Positive primes that satisfy the congruence also stay in prime. Such a prime number is called inert .${\ displaystyle p \ in \ mathbb {Z}}$${\ displaystyle p \ equiv 2 {\ pmod {3}}}$${\ displaystyle \ mathbb {Z} [\ omega]}$
• Positive prime numbers that satisfy the congruence become products of two complex conjugate prime elements. It is said that such prime numbers are decomposed .${\ displaystyle p \ in \ mathbb {Z}}$${\ displaystyle p \ equiv 1 {\ pmod {3}}}$${\ displaystyle \ mathbb {Z} [\ omega]}$

So the inert prime numbers are and a prime factorization of the first decomposed prime numbers is: ${\ displaystyle 2,5,11,17,23, \ ldots}$

${\ displaystyle 7 = (3+ \ omega) \ cdot (2- \ omega), \ quad 13 = (4+ \ omega) \ cdot (3- \ omega), \ quad 19 = (3-2 \ omega) \ cdot (5 + 2 \ omega), \ ldots}$

The six elements associated with a prime element are prime, as is the element complexly conjugated to a prime element . ${\ displaystyle \ alpha}$ ${\ displaystyle {\ overline {\ alpha}}}$

Since the norm of an element of always lies in, the inert whole prime numbers and the prime elements, which appear as factors in the decomposition of the decomposed whole prime numbers, together with their associates, form the set of all prime elements in . ${\ displaystyle N (\ alpha) = \ alpha \ cdot {\ overline {\ alpha}}}$${\ displaystyle \ mathbb {Z} [\ omega]}$${\ displaystyle \ mathbb {Z}}$${\ displaystyle 1- \ omega}$${\ displaystyle \ mathbb {Z} [\ omega]}$

The ring of Eisenstein numbers is Euclidean .

Cubic remainder character

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

If and are a prime element that does not divide, then: ${\ displaystyle \ alpha, \ rho \ in \ mathbb {Z} [\ omega]}$${\ displaystyle \ rho}$${\ displaystyle \ alpha}$

${\ displaystyle \ alpha ^ {N (\ rho) -1} \ equiv 1 {\ pmod {\ rho}}}$

If it holds for the norm of that and so is, then is a power with an integer exponent and we have: ${\ displaystyle \ rho}$${\ displaystyle N (\ rho) \ neq 3}$${\ displaystyle N (\ rho) \ equiv 1 {\ pmod {3}}}$${\ displaystyle \ alpha ^ {\ frac {N (\ rho) -1} {3}}}$

${\ displaystyle \ alpha ^ {\ frac {N (\ rho) -1} {3}} \ equiv \ omega ^ {k} {\ pmod {\ rho}}}$ for a uniquely determined 3rd root of unity ${\ displaystyle \ omega ^ {k}}$

This root of unity is called the cubic residual character of modulo and is written for it: ${\ displaystyle \ alpha}$${\ displaystyle \ rho}$

${\ displaystyle \ left ({\ frac {\ alpha} {\ rho}} \ right) _ {3} = \ omega ^ {k} \ equiv \ alpha ^ {\ frac {N (\ rho) -1} { 3}} {\ pmod {\ rho}}}$

The designation as character results from the fact that the mapping with a fixed prime element determines a unitary character on the multiplicative group of the finite field . ${\ displaystyle \ rho}$${\ displaystyle \ mathbb {Z} [\ omega] / (\ rho)}$

The congruence is solvable if and only if applies. If the congruence is solvable and , then a cubic remainder is called modulo ; the congruence is insoluble, a cubic non-remainder modulo . Likewise, the terms cubic remainder and non-remainder are explained in a more general way if it is coprime to but not a prime element. ${\ displaystyle x ^ {3} \ equiv \ alpha {\ pmod {\ rho}}, \; (\ alpha \ not \ equiv 0 {\ pmod {\ rho}})}$${\ displaystyle Z [\ omega]}$${\ displaystyle \ left ({\ frac {\ alpha} {\ rho}} \ right) _ {3} = 1}$${\ displaystyle \ alpha \ not \ equiv 0 {\ pmod {\ rho}}}$${\ displaystyle \ alpha}$${\ displaystyle \ rho}$${\ displaystyle \ rho}$${\ displaystyle \ rho}$${\ displaystyle \ alpha}$

The cubic remainder character has formal properties for prime elements that are not too associated, which are similar to the properties of the Legendre symbol : ${\ displaystyle \ rho}$${\ displaystyle 1- \ omega}$

1. ${\ displaystyle \ left ({\ frac {\ alpha \ beta} {\ rho}} \ right) _ {3} = \ left ({\ frac {\ alpha} {\ rho}} \ right) _ {3} \ left ({\ frac {\ beta} {\ rho}} \ right) _ {3}}$
2. ${\ displaystyle {\ overline {\ left ({\ frac {\ alpha} {\ rho}} \ right) _ {3}}} = \ left ({\ frac {\ overline {\ alpha}} {\ overline { \ rho}}} \ right) _ {3}}$, where the overline stands for the complex conjugation.
3. If and are associated prime elements, then holds .${\ displaystyle \ rho}$${\ displaystyle \ theta}$${\ displaystyle \ left ({\ frac {\ alpha} {\ rho}} \ right) _ {3} = \ left ({\ frac {\ alpha} {\ theta}} \ right) _ {3}}$
4. Is , then applies .${\ displaystyle \ alpha \ equiv \ beta {\ pmod {\ rho}}}$${\ displaystyle \ left ({\ frac {\ alpha} {\ rho}} \ right) _ {3} = \ left ({\ frac {\ beta} {\ rho}} \ right) _ {3}}$

The cubic remainder can be continued in the “denominator” multiplicatively to composite numbers that are coprime to 3. It is then additionally defined that the residual cubic symbol defined in this way has the value 0 if the numbers in the ring of Eisenstein numbers are not coprime to one another, but are coprime to 3. This generalization is analogous to the generalization of the Legendre symbol to the Jacobi symbol, except for the fact that no value is defined for the symbol in the event that applies or equivalently that the norm of in is divided by 3. Sometimes the symbol 0 is set in the latter case. This variant does not change anything in the following statements. ${\ displaystyle \ left ({\ frac {\ alpha} {\ lambda}} \ right) _ {3}}$${\ displaystyle \ alpha, \ lambda}$${\ displaystyle \ lambda}$${\ displaystyle \ lambda \ equiv 0 {\ pmod {1- \ omega}}}$${\ displaystyle \ lambda}$${\ displaystyle \ mathbb {Z}}$

Similar to the Jacobi symbol , the following statements apply to a “denominator” of the remaining cubic symbol that is not a prime element: ${\ displaystyle \ lambda}$

• By the multiplicative continuation, according to the definition:
${\ displaystyle \ left ({\ frac {\ alpha} {\ lambda}} \ right) _ {3} = \ left ({\ frac {\ alpha} {\ pi _ {1}}} \ right) _ { 3} ^ {\ nu _ {1}} \ left ({\ frac {\ alpha} {\ pi _ {2}}} \ right) _ {3} ^ {\ nu _ {2}} \ cdots \; , \,}$it has a decomposition of different prime elements into pairwise , none of which is to be associated.${\ displaystyle \ lambda = \ pi _ {1} ^ {\ nu _ {1}} \ pi _ {2} ^ {\ nu _ {2}} \ pi _ {3} ^ {\ nu _ {3} } \ dots}$${\ displaystyle \ lambda}$${\ displaystyle \ pi _ {j}}$${\ displaystyle 1- \ omega}$
• If the “numerator” is a cubic remainder modulo and , then the symbol takes the value 1.${\ displaystyle \ alpha}$${\ displaystyle \ lambda}$${\ displaystyle \ lambda \ not \ equiv 0 {\ pmod {1- \ omega}}}$
• If the symbol takes on a value other than 1, then the numerator is not a cubic remainder modulo or not prime to 3.${\ displaystyle \ lambda}$${\ displaystyle \ lambda}$
• The symbol can assume the value 1, even if the numerator is a cubic non-remainder modulo .${\ displaystyle \ lambda}$

Primary numbers

To formulate a cubic reciprocity law on the ring of Eisenstein numbers, certain representatives have to be selected from the associates of an Eisenstein number. Eisenstein calls a number primary if it fulfills the congruence . One can easily prove that for numbers whose norm (in ) is relatively prime to 3, exactly one element associated with them is primary in the sense of this definition. A disadvantage of the definition is that the product of two primary numbers is always the opposite of a primary number. ${\ displaystyle \ lambda}$ ${\ displaystyle \ lambda \ equiv 2 {\ pmod {3}}}$${\ displaystyle \ mathbb {Z}}$

Today one therefore usually defines:

• An Eisenstein number is primary if it is relatively prime to 3 and modulo congruent to an ordinary integer.${\ displaystyle \ lambda}$${\ displaystyle (1- \ omega) ^ {2} = - 3 \ omega}$

This definition is synonymous with the fact that the congruence in the ring of Eisenstein numbers applies. The following then applies: ${\ displaystyle \ lambda \ equiv \ pm 1 {\ pmod {3}}}$

1. If the norm is from coprime to 3, then exactly one of the numbers is primary.${\ displaystyle \ lambda \ in \ mathbb {Z} [\ omega] ^ {*}}$${\ displaystyle \ lambda, \ omega \ cdot \ lambda, \ omega ^ {2} \ cdot \ lambda}$
2. The product of two primary numbers is primary.
3. With every number, the complex number conjugated to it is also primary.
4. A primary number in the modern sense is either itself primary in the Eisenstein sense or it is.${\ displaystyle \ lambda}$${\ displaystyle - \ lambda}$
5. There are always exactly two primary numbers among the associates of a number that is relatively prime to 3 .${\ displaystyle \ pm \ lambda}$

Since −1 is always a cubic remainder, the uniqueness of this definition "except for the sign" is sufficient for the formulation of the reciprocity law.

Cubic reciprocity law

The following applies to two primary numbers : ${\ displaystyle \ alpha, \ beta}$

${\ displaystyle \ left ({\ frac {\ alpha} {\ beta}} \ right) _ {3} = \ left ({\ frac {\ beta} {\ alpha}} \ right) _ {3}}$

For this cubic reciprocity law there are supplementary clauses for the units and the prime element : ${\ displaystyle 1- \ omega}$

If is and holds, then also holds ${\ displaystyle \ lambda = a + b \ omega, (a, b \ in \ mathbb {Z})}$${\ displaystyle a = 3m + 1, b = 3n, (m, n \ in \ mathbb {Z})}$

${\ displaystyle \ left ({\ frac {\ omega} {\ lambda}} \ right) _ {3} = \ omega ^ {\ frac {1-ab} {3}} = \ omega ^ {- mn}, \; \; \; \ left ({\ frac {1- \ omega} {\ lambda}} \ right) _ {3} = \ omega ^ {\ frac {a-1} {3}} = \ omega ^ {m}, \; \; \; \ left ({\ frac {3} {\ lambda}} \ right) _ {3} = \ omega ^ {\ frac {b} {3}} = \ omega ^ { n}.}$

For primary "denominators" with can be replaced by the associated primary element without changing the value of the symbol. ${\ displaystyle \ lambda}$${\ displaystyle a \ equiv 2 {\ pmod {3}}}$${\ displaystyle \ lambda}$${\ displaystyle - \ lambda}$

literature

• David A. Cox: Primes of the form x 2 + ny 2 . Fermat, class field theory and complex multiplication. Wiley, New York 1989, ISBN 0-471-50654-0 .
• Ferdinand Gotthold Eisenstein: Proof of the reciprocity theorem for the cubic remainders in the theory of numbers assembled from the third roots of the unit . In: August Leopold Crelle (ed.): Journal for pure and applied mathematics . No. 27 . Georg Reimer, Berlin 1844, p. 289-310 .
• Kenneth Ireland, Michael Rosen (mathematicians) : A Classical Introduction to Modern Number Theory . 2nd Edition. Springer, New York 1990, ISBN 978-1-4419-3094-1 .
• Franz Lemmermeyer : Reciprocity Laws: From Euler to Eisenstein . Springer, Berlin / Heidelberg / New York / Barcelona / Hong Kong / London / Milan / Paris / Singapore / Tokyo 2000, ISBN 3-540-66957-4 .
• Armin Leutbecher: Number Theory: An Introduction to Algebra . Springer, Berlin / Heidelberg / Singapore / Tokyo / New York / Barcelona / Budapest / Hong Kong / London / Milan / Paris / Santa Clara 1996, ISBN 3-540-58791-8 .