Mathematics Foundations/22.2 Algebraic Number Theory

22.2 Algebraic Number Theory Introduction Algebraic number theory studies algebraic numbers—roots of polynomials with integer coefficients—and their properties. It extends the concepts of elementary number theory to larger number systems, providing powerful tools for solving classical problems and developing modern cryptography. Algebraic Numbers and Integers Definition 22.2.1: An algebraic number is a complex number that is a root of a non-zero polynomial with rational coefficients. Definition 22.2.2: An algebraic integer is a complex number that is a root of a monic polynomial (leading coefficient 1) with integer coefficients. Examples: is an algebraic integer, as it satisfies is an algebraic number but not an algebraic integer is an algebraic integer, as it satisfies Theorem 22.2.3: The set of algebraic integers forms a ring, denoted by . Number Fields Definition 22.2.4: A number field is a finite extension of , i.e., where each is algebraic over . Definition 22.2.5: The ring of integers of a number field , denoted , is the set of all algebraic integers contained in . Theorem 22.2.6: If is a number field of degree over , then is a free -module of rank . Definition 22.2.7: An integral basis for is a set such that every element of can be uniquely written as with . Example 22.2.8: For where is a square-free integer: If , then is an integral basis for If , then is an integral basis for Ideals and Factorization Definition 22.2.9: An ideal of is a non-empty subset such that: If , then If and , then Definition 22.2.10: A prime ideal is a proper ideal such that if , then either or . Theorem 22.2.11: In , every non-zero ideal can be factored uniquely as a product of prime ideals. Definition 22.2.12: Two ideals and are coprime if . Theorem 22.2.13 (Chinese Remainder Theorem for Ideals): If are pairwise coprime ideals in , then: Ideal Class Group Definition 22.2.14: Two ideals and are equivalent, denoted , if there exist non-zero such that . Definition 22.2.15: The ideal class group is the group of equivalence classes of non-zero ideals of under this relation. Theorem 22.2.16: The ideal class group is finite. Its order is called the class number of . Theorem 22.2.17: if and only if is a principal ideal domain (PID). Discriminants and Ramification Definition 22.2.18: The discriminant of a number field with integral basis is: Definition 22.2.19: Let be a number field and a prime number. The factorization of has the form: If some , we say is ramified in If and , we say is inert in If and all , we say is completely split in Theorem 22.2.20: A prime ramifies in if and only if divides the discriminant . Dedekind Zeta Function Definition 22.2.21: The Dedekind zeta function of a number field is defined for by: where the sum is over all non-zero ideals of , and is the norm of the ideal. Theorem 22.2.22: The Dedekind zeta function has an Euler product: where the product is over all prime ideals of . Theorem 22.2.23: The Dedekind zeta function has a simple pole at with residue related to important invariants of : where: is the number of real embeddings of is the number of pairs of complex embeddings is the class number is the regulator is the number of roots of unity in Units and Dirichlet's Unit Theorem Definition 22.2.24: A unit in is an element such that . Theorem 22.2.25 (Dirichlet's Unit Theorem): The group of units is isomorphic to: where is the finite cyclic group of roots of unity in . Definition 22.2.26: A set of units is a fundamental system of units if every unit can be uniquely expressed as: where is a root of unity in and . Local Fields and Completions Definition 22.2.27: For a prime ideal in , the p-adic valuation on is defined by: if Definition 22.2.28: The completion of with respect to is denoted . Theorem 22.2.29: If lies above the rational prime , then is a finite extension of . Class Field Theory Definition 22.2.30: A ray class field of modulo an ideal is the maximal abelian extension of in which all primes in ramify completely. Theorem 22.2.31 (Artin Reciprocity): There exists a canonical isomorphism: where is the group of fractional ideals coprime to and is the subgroup of principal ideals with generators congruent to 1 modulo . Theorem 22.2.32 (Kronecker-Weber): Every finite abelian extension of is contained in a cyclotomic field for some . Applications to Diophantine Equations Example 22.2.33: Fermat's Last Theorem can be approached using algebraic number theory. For the case , if had non-trivial solutions, this would lead to a contradiction in the unique factorization of ideals in where is a primitive cube root of unity. Exercises Exercise 22.2.1: Determine the ring of integers of and find a fundamental unit. Exercise 22.2.2: Factor the ideal in into prime ideals. Exercise 22.2.3: Compute the class number of . Exercise 22.2.4: Prove that if , then remains prime in . Exercise 22.2.5: Show that is not a unique factorization domain by finding two different factorizations of 6.