๐ขAlgebraic Number Theory Unit 2 โ Number Fields and Integer Rings
Number fields are extensions of rational numbers, forming the foundation of algebraic number theory. They're characterized by their degree and contain algebraic numbers, which are roots of polynomials with rational coefficients. These fields play a crucial role in understanding the structure of algebraic integers.
Integer rings in number fields consist of algebraic integers, which are roots of monic polynomials with integer coefficients. These rings have unique properties, including being Dedekind domains, and their study involves concepts like integral bases, discriminants, and fractional ideals. Understanding these structures is essential for deeper number theory exploration.
Study Guides for Unit 2 โ Number Fields and Integer Rings
Number field K is a finite extension of the field of rational numbers Q
Degree of a number field [K:Q] is the dimension of K as a vector space over Q
Algebraic number is a complex number that is a root of a non-zero polynomial with rational coefficients
Algebraic integer is an algebraic number that is a root of a monic polynomial with integer coefficients
Ring of integers OKโ of a number field K consists of all algebraic integers in K
Example: In the number field Q(5โ), the ring of integers is Z[21+5โโ]
Ideal in a ring R is a subset IโR closed under addition and absorption by ring elements
Principal ideal is an ideal generated by a single element aโR, denoted as (a)={ra:rโR}
Unique factorization domain (UFD) is an integral domain where every non-zero element can be uniquely factored into irreducible elements up to ordering and unit factors
Number Fields: Foundation and Properties
Number field K is a field extension of Q of finite degree n=[K:Q]
Every number field K can be obtained by adjoining a single algebraic number ฮฑ to Q, i.e., K=Q(ฮฑ)
Minimal polynomial of an algebraic number ฮฑ is the monic irreducible polynomial mฮฑโ(x)โQ[x] of lowest degree such that mฮฑโ(ฮฑ)=0
Degree of ฮฑ is the degree of its minimal polynomial
Primitive element theorem states that every number field K has a primitive element ฮฑ such that K=Q(ฮฑ)
Embeddings of a number field K into C are field homomorphisms ฯ:KโC fixing Q
Number of embeddings equals the degree of the number field
Trace and norm of an element ฮฑโK are defined as:
TrK/Qโ(ฮฑ)=โฯโฯ(ฮฑ)
NK/Qโ(ฮฑ)=โฯโฯ(ฮฑ)
where ฯ runs through all embeddings of K into C
Integer Rings in Number Fields
Ring of integers OKโ of a number field K is the set of all algebraic integers in K
OKโ is a subring of K and a free Z-module of rank equal to the degree of K
Integral basis of OKโ is a basis of OKโ as a Z-module
Example: In Q(2โ), an integral basis is {1,2โ}
Discriminant of a number field K is defined as ฮKโ=det(TrK/Qโ(ฮฑiโฮฑjโ))1โคi,jโคnโ, where {ฮฑ1โ,โฆ,ฮฑnโ} is an integral basis of OKโ
Dedekind domain is an integral domain where every ideal can be uniquely factored into prime ideals
Ring of integers OKโ is a Dedekind domain
Fractional ideal of OKโ is a non-zero OKโ-submodule of K
Every fractional ideal is invertible, i.e., for every fractional ideal I, there exists a fractional ideal J such that IJ=OKโ
Algebraic Integers and Their Properties
Algebraic integer is an algebraic number that satisfies a monic polynomial with integer coefficients
Sum, difference, and product of algebraic integers are also algebraic integers
Every algebraic integer belongs to the ring of integers OKโ of some number field K
Norm and trace of an algebraic integer are integers
For ฮฑโOKโ, TrK/Qโ(ฮฑ)โZ and NK/Qโ(ฮฑ)โZ
Algebraic integers in a number field K form a ring, which is the ring of integers OKโ
Units in OKโ are algebraic integers with norm ยฑ1
Example: In Z[2โ], the units are ยฑ1
Dirichlet's unit theorem states that the group of units in OKโ is finitely generated
Ideals in Integer Rings
Ideal in the ring of integers OKโ is an additive subgroup closed under multiplication by elements of OKโ
Principal ideal in OKโ is an ideal generated by a single element ฮฑโOKโ, denoted as (ฮฑ)={ฮฑฮฒ:ฮฒโOKโ}
Prime ideal p in OKโ is a proper ideal such that for any ฮฑ,ฮฒโOKโ, if ฮฑฮฒโp, then either ฮฑโp or ฮฒโp
Maximal ideal in OKโ is a proper ideal that is not contained in any other proper ideal
Every maximal ideal is prime, but not every prime ideal is maximal
Norm of an ideal I in OKโ is defined as N(I)=โฃOKโ/Iโฃ, the cardinality of the quotient ring
For a principal ideal (ฮฑ), N((ฮฑ))=โฃNK/Qโ(ฮฑ)โฃ
Sum and product of ideals in OKโ are also ideals in OKโ
Factorization and Unique Factorization Domains
Unique factorization domain (UFD) is an integral domain where every non-zero element can be uniquely factored into irreducible elements up to ordering and unit factors
Ring of integers OKโ is a UFD if and only if every ideal in OKโ is principal
Example: Z[โ5โ] is not a UFD, as the ideal (2,1+โ5โ) is not principal
In a Dedekind domain, every ideal can be uniquely factored into prime ideals
Ring of integers OKโ is a Dedekind domain, so every ideal in OKโ has a unique factorization into prime ideals
Class number of a number field K is the cardinality of the ideal class group, which measures the failure of unique factorization in OKโ
K has class number 1 if and only if OKโ is a UFD
Euclidean domain is an integral domain with a Euclidean function, which implies it is a UFD
Example: Z[2โ] is a Euclidean domain with the Euclidean function N(a+b2โ)=โฃa2โ2b2โฃ
Applications and Examples
Number fields and their rings of integers have applications in cryptography, such as in the construction of lattice-based cryptosystems
Example: The ring of integers of a cyclotomic field can be used to construct a lattice for the NTRU cryptosystem
Algebraic number theory is used in the study of Diophantine equations, which are polynomial equations with integer coefficients and integer solutions
Example: Fermat's Last Theorem states that the equation xn+yn=zn has no non-zero integer solutions for n>2
Class number formula relates the class number of a number field to its discriminant and the values of its Dedekind zeta function
Example: For a quadratic field K=Q(dโ), the class number hKโ satisfies hKโ=2ฯโฃฮKโโฃโโL(1,ฯdโ), where L(s,ฯdโ) is the Dirichlet L-function associated with the quadratic character ฯdโ
Primes in Z may factor into prime ideals in the ring of integers of a number field
Example: In Z[โ5โ], the prime 2 factors as (2)=(2,1+โ5โ)(2,1โโ5โ)
Units in the ring of integers form a finitely generated abelian group, as described by Dirichlet's unit theorem
Example: In Q(2โ), the units are ยฑ(1+2โ)n for nโZ
Advanced Topics and Extensions
Dedekind zeta function of a number field K is a generalization of the Riemann zeta function, defined as ฮถKโ(s)=โaโN(a)s1โ, where a runs through all non-zero ideals of OKโ
Analytic class number formula expresses the residue of ฮถKโ(s) at s=1 in terms of the class number, regulator, and other invariants of K
Artin reciprocity law is a central result in class field theory, relating the abelian extensions of a number field to its idele class group
Ideles of a number field K are elements of the restricted product of the completions of K with respect to its non-archimedean absolute values
Langlands program is a vast network of conjectures connecting representation theory, automorphic forms, and arithmetic geometry
Langlands reciprocity conjecture relates Galois representations to automorphic representations of reductive groups over number fields
Elliptic curves over number fields have a rich arithmetic structure and are connected to various problems in number theory
Mordell-Weil theorem states that the group of K-rational points on an elliptic curve over a number field K is finitely generated
Iwasawa theory studies the growth of arithmetic objects (such as class groups) in towers of number fields
Main conjecture of Iwasawa theory relates the characteristic ideal of the Iwasawa module to the p-adic L-function
Stark conjectures provide a conjectural description of the leading term of the Taylor expansion of Artin L-functions at s=0 in terms of units in number fields