In mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle \mathbb {Q} } such that the field extension K / Q {\displaystyle K/\mathbb {Q} } has finite degree (and hence is an algebraic field extension). Thus K {\displaystyle K} is a field that contains Q {\displaystyle \mathbb {Q} } and has finite dimension when considered as a vector space over Q {\displaystyle \mathbb {Q} } . The study of algebraic number fields, that is, of algebraic extensions of the field of rational numbers, is the central topic of algebraic number theory. This study reveals hidden structures behind the rational numbers, by using algebraic methods.
Definition
Prerequisites
The notion of algebraic number field relies on the concept of a field. A field consists of a set of elements together with two operations, namely addition, and multiplication, and some distributivity assumptions. These operations make the field into an abelian group under addition, and they make the nonzero elements of the field into another abelian group under multiplication. A prominent example of a field is the field of rational numbers, commonly denoted Q {\displaystyle \mathbb {Q} } , together with its usual operations of addition and multiplication. Another notion needed to define algebraic number fields is vector spaces. To the extent needed here, vector spaces can be thought of as consisting of sequences (or tuples)
( x 1 , x 2 , … ) {\displaystyle (x_{1},x_{2},\dots )}
whose entries are elements of a fixed field, such as the field Q {\displaystyle \mathbb {Q} } . Any two such sequences can be added by adding the corresponding entries. Furthermore, all members of any sequence can be multiplied by a single element c of the fixed field. These two operations known as vector addition and scalar multiplication satisfy a number of properties that serve to define vector spaces abstractly. Vector spaces are allowed to be "infinite-dimensional", that is to say that the sequences constituting the vector spaces may be of infinite length. If, however, the vector space consists of finite sequences
( x 1 , … , x n ) {\displaystyle (x_{1},\dots ,x_{n})} , the vector space is said to be of finite dimension, n {\displaystyle n} .
Definition An algebraic number field (or simply number field) is a finite-degree field extension of the field of rational numbers. Here degree means the dimension of the field as a vector space over Q {\displaystyle \mathbb {Q} } .
Examples The smallest and most basic number field is the field Q {\displaystyle \mathbb {Q} } of rational numbers. Many properties of general number fields are modeled after the properties of Q {\displaystyle \mathbb {Q} } . At the same time, many other properties of algebraic number fields are substantially different from the properties of rational numbers—one notable example is that the ring of algebraic integers of a number field is not necessarily a principal ideal domain, and not necessarily even a unique factorization domain. The Gaussian rationals, denoted Q ( i ) {\displaystyle \mathbb {Q} (i)} (read as " Q {\displaystyle \mathbb {Q} } adjoin i {\displaystyle i} "), form the first (historically) non-trivial example of a number field. Its elements are elements of the form a + b i {\displaystyle a+bi} where both a {\displaystyle a} and b {\displaystyle b} are rational numbers and i {\displaystyle i} is the imaginary unit. Such expressions may be added, subtracted, and multiplied according to the usual rules of arithmetic and then simplified using the identity i 2 = − 1 {\displaystyle i^{2}=-1} . Explicitly, for real numbers a , b , c , d {\displaystyle a,b,c,d} :
( a + b i ) + ( c + d i ) = ( a + c ) + ( b + d ) i ( a + b i ) ⋅ ( c + d i ) = ( a c − b d ) + ( a d + b c ) i {\displaystyle {\begin{aligned}&(a+bi)+(c+di)=(a+c)+(b+d)i\\&(a+bi)\cdot (c+di)=(ac-bd)+(ad+bc)i\end{aligned}}}
Non-zero Gaussian rational numbers are invertible, which can be seen from the identity
( a + b i ) ( a a 2 + b 2 − b a 2 + b 2 i ) = ( a + b i ) ( a − b i ) a 2 + b 2 = 1. {\displaystyle (a+bi)\left({\frac {a}{a^{2}+b^{2}}}-{\frac {b}{a^{2}+b^{2}}}i\right)={\frac {(a+bi)(a-bi)}{a^{2}+b^{2}}}=1.}
It follows that the Gaussian rationals form a number field that is two-dimensional as a vector space over Q {\displaystyle \mathbb {Q} } . More generally, for any square-free integer d {\displaystyle d} , the quadratic field Q ( d ) {\displaystyle \mathbb {Q} ({\sqrt {d}})} is a number field obtained by adjoining the square root of d {\displaystyle d} to the field of rational numbers. Arithmetic operations in this field are defined in analogy with the case of Gaussian rational numbers, d = − 1 {\displaystyle d=-1} . The cyclotomic field Q ( ζ n ) , {\displaystyle \mathbb {Q} (\zeta _{n}),} where ζ n = exp ( 2 π i / n ) {\displaystyle \zeta _{n}=\exp {(2\pi i/n)}} , is a number field obtained from Q {\displaystyle \mathbb {Q} } by adjoining a primitive n-th root of unity ζ n {\displaystyle \zeta _{n}} . This field contains all complex nth roots of unity and its dimension over Q {\displaystyle \mathbb {Q} } is equal to φ ( n ) {\displaystyle \varphi (n)} , where φ {\displaystyle \varphi } is the Euler totient function.
Non-examples The real numbers, R {\displaystyle \mathbb {R} } , and the complex numbers, C {\displaystyle \mathbb {C} } , are fields that have infinite dimension as Q {\displaystyle \mathbb {Q} } -vector spaces; hence, they are not number fields. This follows from the uncountability of R {\displaystyle \mathbb {R} } and C {\displaystyle \mathbb {C} } as sets, whereas every number field is necessarily countable, as they are finite-dimensional vector spaces over Q {\displaystyle \mathbb {Q} } . The set Q 2 {\displaystyle \mathbb {Q} ^{2}} of ordered pairs of rational numbers, with the entry-wise addition and multiplication is a two-dimensional commutative algebra over Q {\displaystyle \mathbb {Q} } . However, it is not a field, since it has zero divisors: ( 1 , 0 ) ⋅ ( 0 , 1 ) = ( 0 , 0 ) {\displaystyle (1,0)\cdot (0,1)=(0,0)} .
Algebraicity, and ring of integers Generally, in abstract algebra, a field extension K / L {\displaystyle K/L} is algebraic if every element f {\displaystyle f} of the bigger field K {\displaystyle K} is the zero of a (nonzero) polynomial with coefficients e 0 , … , e m {\displaystyle e_{0},\ldots ,e_{m}} in L {\displaystyle L} :
p ( f ) = e m f m + e m − 1 f m − 1 + ⋯ + e 1 f + e 0 = 0 {\displaystyle p(f)=e_{m}f^{m}+e_{m-1}f^{m-1}+\cdots +e_{1}f+e_{0}=0}
Every field extension of finite degree is algebraic. (Proof: for x {\displaystyle x} in K {\displaystyle K} , simply consider 1 , x , x 2 , x 3 , … {\displaystyle 1,x,x^{2},x^{3},\ldots } – we get a linear dependence, i.e. a polynomial that x {\displaystyle x} is a root of.) In particular this applies to algebraic number fields, so any element f {\displaystyle f} of an algebraic number field K {\displaystyle K} can be written as a zero of a polynomial with rational coefficients. Therefore, elements of K {\displaystyle K} are also referred to as algebraic numbers. Given a polynomial p {\displaystyle p} such that p ( f ) = 0 {\displaystyle p(f)=0} , it can be arranged such that the leading coefficient e m {\displaystyle e_{m}} is one, by dividing all coefficients by it, if necessary. A polynomial with this property is known as a monic polynomial. In general it will have rational coefficients. If, however, the monic polynomial's coefficients are actually all integers, f {\displaystyle f} is called an algebraic integer. Any (usual) integer z ∈ Z {\displaystyle z\in \mathbb {Z} } is an algebraic integer, as it is the zero of the linear monic polynomial:
p ( t ) = t − z {\displaystyle p(t)=t-z} . It can be shown that any algebraic integer that is also a rational number must actually be an integer, hence the name "algebraic integer". Again using abstract algebra, specifically the notion of a finitely generated module, it can be shown that the sum and the product of any two algebraic integers is still an algebraic integer. It follows that the algebraic integers in K {\displaystyle K} form a ring denoted O K {\displaystyle {\mathcal {O}}_{K}} called the ring of integers of K {\displaystyle K} . It is a subring of (that is, a ring contained in) K {\displaystyle K} . A field contains no zero divisors and this property is inherited by any subring, so the ring of integers of K {\displaystyle K} is an integral domain. The field K {\displaystyle K} is the field of fractions of the integral domain O K {\displaystyle {\mathcal {O}}_{K}} . This way one can get back and forth between the algebraic number field K {\displaystyle K} and its ring of integers O K {\displaystyle {\mathcal {O}}_{K}} . Rings of algebraic integers have three distinctive properties: firstly, O K {\displaystyle {\mathcal {O}}_{K}} is an integral domain that is integrally closed in its field of fractions K {\displaystyle K} . Secondly, O K {\displaystyle {\mathcal {O}}_{K}} is a Noetherian ring. Finally, every nonzero prime ideal of O K {\displaystyle {\mathcal {O}}_{K}} is maximal or, equivalently, the Krull dimension of this ring is one. An abstract commutative ring with these three properties is called a Dedekind ring (or Dedekind domain), in honor of Richard Dedekind, who undertook a deep study of rings of algebraic integers.
Unique factorization For general Dedekind rings, in particular rings of integers, there is a unique factorization of ideals into a product of prime ideals. For example, the ideal ( 6 ) {\displaystyle (6)} in the ring Z [ − 5 ] {\displaystyle \mathbf {Z} [{\sqrt {-5}}]} of quadratic integers factors into prime ideals as
( 6 ) = ( 2 , 1 + − 5 ) ( 2 , 1 − − 5 ) ( 3 , 1 + − 5 ) ( 3 , 1 − − 5 ) {\displaystyle (6)=(2,1+{\sqrt {-5}})(2,1-{\sqrt {-5}})(3,1+{\sqrt {-5}})(3,1-{\sqrt {-5}})}
However, unlike Z {\displaystyle \mathbf {Z} } as the ring of integers of Q {\displaystyle \mathbf {Q} } , the ring of integers of a proper extension of Q {\displaystyle \mathbf {Q} } need not admit unique factorization of numbers into a product of prime numbers or, more precisely, prime elements. This happens already for quadratic integers, for example in O Q ( − 5 ) = Z [ − 5 ] {\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}})}=\mathbf {Z} [{\sqrt {-5}}]} , the uniqueness of the factorization fails:
6 = 2 ⋅ 3 = ( 1 + − 5 ) ⋅ ( 1 − − 5 ) {\displaystyle 6=2\cdot 3=(1+{\sqrt {-5}})\cdot (1-{\sqrt {-5}})}
Using the norm it can be shown that these two factorization are actually inequivalent in the sense that the factors do not just differ by a unit in O Q ( − 5 ) {\displaystyle {\mathcal {O}}_{\mathbf {Q} ({\sqrt {-5}})}} . Euclidean domains are unique factorization domains: For example Z [ i ] {\displaystyle \mathbf {Z} [i]} , the ring of Gaussian integers, and Z [ ω ] {\displaystyle \mathbf {Z} [\omega ]} , the ring of Eisenstein integers, where ω {\displaystyle \omega } is a cube root of unity (unequal to 1), have this property.
Analytic objects: ζ-functions, L-functions, and class number formula The failure of unique factorization is measured by the class number, commonly denoted h, the cardinality of the so-called ideal class group. This group is always finite. The ring of integers O K {\displaystyle {\mathcal {O}}_{K}} possesses unique factorization if and only if it is a principal ring or, equivalently, if K {\displaystyle K} has class number 1. Given a number field, the class number is often difficult to compute. The class number problem, going back to Gauss, is concerned with the existence of imaginary quadratic number fields (i.e., Q ( − d ) , d ≥ 1 {\displaystyle \mathbf {Q} ({\sqrt {-d}}),d\geq 1} ) with prescribed class number. The class number formula relates h to other fundamental invariants of K {\displaystyle K} . It involves the Dedekind zeta function ζ K ( s ) {\displaystyle \zeta _{K}(s)} , a function in a complex variable s {\displaystyle s} , defined by
ζ K ( s ) := ∏ p 1 1 − N ( p ) − s . {\displaystyle \zeta _{K}(s):=\prod _{\mathfrak {p}}{\frac {1}{1-N({\mathfrak {p}})^{-s}}}.}
(The product is over all prime ideals of O K {\displaystyle {\mathcal {O}}_{K}} , N ( p ) {\displaystyle N({\mathfrak {p}})} denotes the norm of the prime ideal or, equivalently, the (finite) number of elements in the residue field O K / p {\displaystyle {\mathcal {O}}_{K}/{\mathfrak {p}}} . The infinite product converges only for Re(s) > 1; in general analytic continuation and the functional equation for the zeta-function are needed to define the function for all s). The Dedekind zeta-function generalizes the Riemann zeta-function in that ζ Q {\displaystyle \mathbb {Q} } (s) = ζ(s). The class number formula states that ζ K {\displaystyle K} (s) has a simple pole at s = 1 and at this point the residue is given by
2 r 1 ⋅ ( 2 π ) r 2 ⋅ h ⋅ Reg w ⋅ | D | . {\displaystyle {\frac {2^{r_{1}}\cdot (2\pi )^{r_{2}}\cdot h\cdot \operatorname {Reg} }{w\cdot {\sqrt {|D|}}}}.}
Here r1 and r2 classically denote the number of real embeddings and pairs of complex embeddings of K {\displaystyle K} , respectively. Moreover, Reg is the regulator of K {\displaystyle K} , w the number of roots of unity in K {\displaystyle K} and D is the discriminant of K {\displaystyle K} . Dirichlet L-functions L ( χ , s ) {\displaystyle L(\chi ,s)} are a more refined variant of ζ ( s ) {\displaystyle \zeta (s)} . Both types of functions encode the arithmetic behavior of Q {\displaystyle \mathbb {Q} } and K {\displaystyle K} , respectively. For example, Dirichlet's theorem asserts that in any arithmetic progression
a , a + m , a + 2 m , … {\displaystyle a,a+m,a+2m,\ldots }
with coprime a {\displaystyle a} and m {\displaystyle m} , there are infinitely many prime numbers. This theorem is implied by the fact that the Dirichlet L {\displaystyle L} -function is nonzero at s = 1 {\displaystyle s=1} . Using much more advanced techniques including algebraic K-theory and Tamagawa measures, modern number theory deals with a description, if largely conjectural (see Tamagawa number conjecture), of values of more general L-functions.
Bases for number fields
Integral basis An integral basis for a number field K {\displaystyle K} of degree n {\displaystyle n} is a set
B = {b1, …, bn} of n algebraic integers in K {\displaystyle K} such that every element of the ring of integers O K {\displaystyle {\mathcal {O}}_{K}} of K {\displaystyle K} can be written uniquely as a Z-linear combination of elements of B; that is, for any x in O K {\displaystyle {\mathcal {O}}_{K}} we have
x = m1b1 + ⋯ + mnbn, where the mi are (ordinary) integers. It is then also the case that any element of K {\displaystyle K} can be written uniquely as
m1b1 + ⋯ + mnbn, where now the mi are rational numbers. The algebraic integers of K {\displaystyle K} are then precisely those elements of K {\displaystyle K} where the mi are all integers. Working locally and using tools such as the Frobenius map, it is always possible to explicitly compute such a basis, and it is now standard for computer algebra systems to have built-in programs to do this.
Power basis Let K {\displaystyle K} be a number field of degree n {\displaystyle n} . Among all possible bases of K {\displaystyle K} (seen as a Q {\displaystyle \mathbb {Q} } -vector space), there are particular ones known as power bases, that are bases of the form
B x = { 1 , x , x 2 , … , x n − 1 } {\displaystyle B_{x}=\{1,x,x^{2},\ldots ,x^{n-1}\}}
for some element x ∈ K {\displaystyle x\in K} . By the primitive element theorem, there exists such an x {\displaystyle x} , called a primitive element. If x {\displaystyle x} can be chosen in O K {\displaystyle {\mathcal {O}}_{K}} and such that B x {\displaystyle B_{x}} is a basis of O K {\displaystyle {\mathcal {O}}_{K}} as a free Z-module, then B x {\displaystyle B_{x}} is called a power integral basis, and the field K {\displaystyle K} is called a monogenic field. An example of a number field that is not monogenic was first given by Dedekind. His example is the field obtained by adjoining a root of the polynomial
x 3 − x 2 − 2 x − 8. {\displaystyle x^{3}-x^{2}-2x-8.}
Regular representation, trace and discriminant Recall that any field extension K / Q {\displaystyle K/\mathbb {Q} } has a unique Q {\displaystyle \mathbb {Q} } -vector space structure. Using the multiplication in K {\displaystyle K} , an element x {\displaystyle x} of the field K {\displaystyle K} over the base field Q {\displaystyle \mathbb {Q} } may be represented by n × n {\displaystyle n\times n} matrices
A = A ( x ) = ( a i j ) 1 ≤ i , j ≤ n {\displaystyle A=A(x)=(a_{ij})_{1\leq i,j\leq n}}
by requiring
x e i = ∑ j = 1 n a i j e j , a i j ∈ Q . {\displaystyle xe_{i}=\sum _{j=1}^{n}a_{ij}e_{j},\quad a_{ij}\in \mathbb {Q} .}
Here e 1 , … , e n {\displaystyle e_{1},\ldots ,e_{n}} is a fixed basis for K {\displaystyle K} , viewed as a Q {\displaystyle \mathbb {Q} } -vector space. The rational numbers a i j {\displaystyle a_{ij}} are uniquely determined by x {\displaystyle x} and the choice of a basis since any element of K {\displaystyle K} can be uniquely represented as a linear combination of the basis elements. This way of associating a matrix to any element of the field K {\displaystyle K} is called the regular representation. The square matrix A {\displaystyle A} represents the effect of multiplication by x {\displaystyle x} in the given basis. It follows that if the element y {\displaystyle y} of K {\displaystyle K} is represented by a matrix B {\displaystyle B} , then the product x y {\displaystyle xy} is represented by the matrix product B A {\displaystyle BA} . Invariants of matrices, such as the trace, determinant, and characteristic polynomial, depend solely on the field element x {\displaystyle x} and not on the basis. In particular, the trace of the matrix A ( x ) {\displaystyle A(x)} is called the trace of the field element x {\displaystyle x} and denoted Tr ( x ) {\displaystyle {\text{Tr}}(x)} , and the determinant is called the norm of x and denoted N ( x ) {\displaystyle N(x)} . Now this can be generalized slightly by instead considering a field extension K / L {\displaystyle K/L} and giving an L {\displaystyle L} -basis for K {\displaystyle K} . Then, there is an associated matrix A K / L ( x ) {\displaystyle A_{K/L}(x)} , which has trace Tr K / L ( x ) {\displaystyle {\text{Tr}}_{K/L}(x)} and norm N K / L ( x ) {\displaystyle {\text{N}}_{K/L}(x)} defined as the trace and determinant of the matrix A K / L ( x ) {\displaystyle A_{K/L}(x)} .
Example Consider the field extension Q ( θ ) {\displaystyle \mathbb {Q} (\theta )} with θ = ζ 3 2 3 {\displaystyle \theta =\zeta _{3}{\sqrt[{3}]{2}}} , where ζ 3 {\displaystyle \zeta _{3}} denotes the cube root of unity exp ( 2 π i / 3 ) . {\displaystyle \exp(2\pi i/3).} Then, we have a Q {\displaystyle \mathbb {Q} } -basis given by { 1 , ζ 3 2 3 , ( ζ 3 2 3 ) 2 }