Cyclotomic field
Encyclopedia
In number theory
, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational number
s. The n-th cyclotomic field Q(ζn) (with n > 2) is obtained by adjoining a primitive n-th root of unity ζn to the rational numbers.
The cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's last theorem
. It was in the process of his deep investigations of the arithmetic of these fields (for prime
n) – and more precisely, because of the failure of unique factorization in their rings of integers
– that Ernst Kummer
first introduced the concept of an ideal number
and proved his celebrated congruences.
of the polynomial
and therefore it is a Galois extension
of the field of rational numbers. The degree of the extension
is given by φ(n) where φ is Euler's phi function. A complete set of Galois conjugates is given by { (ζn)a }, where a runs over the set of invertible residues modulo n (so that a is relative prime to n). The Galois group
is naturally isomorphic
to the multiplicative group
×
of invertible residues modulo n, and it acts on the primitive nth roots of unity by the formula
made early inroads in the theory of cyclotomic fields, in connection with the geometrical problem of constructing a regular polygon with a compass and straightedge
. His surprising result that had escaped his predecessors was that a regular heptadecagon
(with 17 sides) could be so constructed. More generally, if p is a prime number, then a regular p-gon can be constructed if and only if p is a Fermat prime. The geometric problem for a general n can be reduced to the following question in Galois theory
: can the nth cyclotomic field be built as a sequence of quadratic extensions?
is to factor the binomial xn + yn,
where n is an odd prime, appearing in one side of Fermat's equation
as follows:
Here x and y are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field Q(ζn). If unique factorization
of algebraic integers were true, then it could have been used to rule out the existence of nontrivial solutions to Fermat's equation.
Several attempts to tackle Fermat's Last Theorem proceeded along these lines, and both Fermat's proof for n = 4 and Euler's proof for n = 3 can be recast in these terms. Unfortunately, the unique factorization fails in general – for example, for n = 23 – but Kummer
found a way around this difficulty. He introduced a replacement for the prime numbers in the cyclotomic field Q(ζp), expressed the failure of unique factorization quantitatively via the class number hp and proved that if hp is not divisible by p (such numbers p are called regular primes) then Fermat's theorem is true for the exponent n = p. Furthermore, he gave a criterion to determine which primes are regular and using it, established Fermat's theorem for all prime exponents p less than 100, with the exception of the irregular primes 37, 59, and 67. Kummer's work on the congruences for the class numbers of cyclotomic fields was generalized in the twentieth century by Iwasawa
in Iwasawa theory
and by Kubota and Leopoldt in their theory of p-adic zeta functions.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to Q, the field of rational number
Rational number
In mathematics, a rational number is any number that can be expressed as the quotient or fraction a/b of two integers, with the denominator b not equal to zero. Since b may be equal to 1, every integer is a rational number...
s. The n-th cyclotomic field Q(ζn) (with n > 2) is obtained by adjoining a primitive n-th root of unity ζn to the rational numbers.
The cyclotomic fields played a crucial role in the development of modern algebra and number theory because of their relation with Fermat's last theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
. It was in the process of his deep investigations of the arithmetic of these fields (for prime
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
n) – and more precisely, because of the failure of unique factorization in their rings of integers
Ring of integers
In mathematics, the ring of integers is the set of integers making an algebraic structure Z with the operations of integer addition, negation, and multiplication...
– that Ernst Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
first introduced the concept of an ideal number
Ideal number
In number theory an ideal number is an algebraic integer which represents an ideal in the ring of integers of a number field; the idea was developed by Ernst Kummer, and led to Richard Dedekind's definition of ideals for rings...
and proved his celebrated congruences.
Properties
A cyclotomic field is the splitting fieldSplitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...
of the polynomial
- xn − 1
and therefore it is a Galois extension
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
of the field of rational numbers. The degree of the extension
- [Q(ζn):Q]
is given by φ(n) where φ is Euler's phi function. A complete set of Galois conjugates is given by { (ζn)a }, where a runs over the set of invertible residues modulo n (so that a is relative prime to n). The Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
is naturally isomorphic
Natural transformation
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...
to the multiplicative group
×
of invertible residues modulo n, and it acts on the primitive nth roots of unity by the formula
- b: (ζn)a → (ζn)ab.
Relation with regular polygons
GaussCarl Friedrich Gauss
Johann Carl Friedrich Gauss was a German mathematician and scientist who contributed significantly to many fields, including number theory, statistics, analysis, differential geometry, geodesy, geophysics, electrostatics, astronomy and optics.Sometimes referred to as the Princeps mathematicorum...
made early inroads in the theory of cyclotomic fields, in connection with the geometrical problem of constructing a regular polygon with a compass and straightedge
Compass and straightedge
Compass-and-straightedge or ruler-and-compass construction is the construction of lengths, angles, and other geometric figures using only an idealized ruler and compass....
. His surprising result that had escaped his predecessors was that a regular heptadecagon
Heptadecagon
In geometry, a heptadecagon is a seventeen-sided polygon.-Heptadecagon construction:The regular heptadecagon is a constructible polygon, as was shown by Carl Friedrich Gauss in 1796 at the age of 19....
(with 17 sides) could be so constructed. More generally, if p is a prime number, then a regular p-gon can be constructed if and only if p is a Fermat prime. The geometric problem for a general n can be reduced to the following question in Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
: can the nth cyclotomic field be built as a sequence of quadratic extensions?
Relation with Fermat's Last Theorem
A natural approach to proving Fermat's Last TheoremFermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
is to factor the binomial xn + yn,
where n is an odd prime, appearing in one side of Fermat's equation
- xn + yn = zn
as follows:
- xn + yn = (x + y) (x + ζy) ... (x + ζn−1y).
Here x and y are ordinary integers, whereas the factors are algebraic integers in the cyclotomic field Q(ζn). If unique factorization
Fundamental theorem of arithmetic
In number theory, the fundamental theorem of arithmetic states that any integer greater than 1 can be written as a unique product of prime numbers...
of algebraic integers were true, then it could have been used to rule out the existence of nontrivial solutions to Fermat's equation.
Several attempts to tackle Fermat's Last Theorem proceeded along these lines, and both Fermat's proof for n = 4 and Euler's proof for n = 3 can be recast in these terms. Unfortunately, the unique factorization fails in general – for example, for n = 23 – but Kummer
Ernst Kummer
Ernst Eduard Kummer was a German mathematician. Skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a gymnasium, the German equivalent of high school, where he inspired the mathematical career of Leopold Kronecker.-Life:Kummer...
found a way around this difficulty. He introduced a replacement for the prime numbers in the cyclotomic field Q(ζp), expressed the failure of unique factorization quantitatively via the class number hp and proved that if hp is not divisible by p (such numbers p are called regular primes) then Fermat's theorem is true for the exponent n = p. Furthermore, he gave a criterion to determine which primes are regular and using it, established Fermat's theorem for all prime exponents p less than 100, with the exception of the irregular primes 37, 59, and 67. Kummer's work on the congruences for the class numbers of cyclotomic fields was generalized in the twentieth century by Iwasawa
Kenkichi Iwasawa
Kenkichi Iwasawa was a Japanese mathematician who is known for his influence on algebraic number theory.Iwasawa was born in Shinshuku-mura, a town near Kiryū, in Gunma Prefecture...
in Iwasawa theory
Iwasawa theory
In number theory, Iwasawa theory is the study of objects of arithmetic interest over infinite towers of number fields. It began as a Galois module theory of ideal class groups, initiated by Kenkichi Iwasawa, in the 1950s, as part of the theory of cyclotomic fields. In the early 1970s, Barry Mazur...
and by Kubota and Leopoldt in their theory of p-adic zeta functions.