Computing Hilbert class fields of quartic fields using complex - PowerPoint PPT Presentation

Computing Hilbert class fields of quartic fields using complex multiplication Jared Asuncion supervisors: Andreas Enge, Marco Streng 28 November 2019 Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 1 / 14

Definition (degree of a field extension) Let L / K be a field extension. The degree [ L : K ] of a field extension L / K is defined to be the dimension of L as a K-vector space. Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Definition (degree of a field extension) Let L / K be a field extension. The degree [ L : K ] of a field extension L / K is defined to be the dimension of L as a K-vector space. For example, consider the field extension Q ( √− 5) / Q . Q ( √− 5) is a Q -vector space with basis { 1 , √− 5 } . Hence [ Q [ √− 5] : Q ] = 2. Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Definition (degree of a field extension) Let L / K be a field extension. The degree [ L : K ] of a field extension L / K is defined to be the dimension of L as a K-vector space. Definition (algebraic number field) An algebraic number field is a field extension of Q of finite degree. Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Definition (degree of a field extension) Let L / K be a field extension. The degree [ L : K ] of a field extension L / K is defined to be the dimension of L as a K-vector space. Definition (algebraic number field) An algebraic number field is a field extension of Q of finite degree. [ Q ( √− 5) : Q ] = 2. Hence it is a number field. Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Definition (degree of a field extension) Let L / K be a field extension. The degree [ L : K ] of a field extension L / K is defined to be the dimension of L as a K-vector space. Definition (algebraic number field) An algebraic number field is a field extension of Q of finite degree. Definition (group of automorphisms of L fixing K ) Aut( L / K ) = { σ ∈ Aut L : σ ( x ) = x for each x ∈ K } σ ∈ Aut L is an invertible ring homomorphism from L to L . Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Definition (degree of a field extension) Let L / K be a field extension. The degree [ L : K ] of a field extension L / K is defined to be the dimension of L as a K-vector space. Definition (algebraic number field) An algebraic number field is a field extension of Q of finite degree. Definition (group of automorphisms of L fixing K ) Aut( L / K ) = { σ ∈ Aut L : σ ( x ) = x for each x ∈ K } σ ∈ Aut L is an invertible ring homomorphism from L to L . √ √ √ √ √ Aut( Q ( − 5) / Q ) = { a + b − 5 �→ a + b − 5 , a + b − 5 �→ a − b − 5 } . It is a group of order 2. Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Definition (degree of a field extension) Let L / K be a field extension. The degree [ L : K ] of a field extension L / K is defined to be the dimension of L as a K-vector space. Definition (algebraic number field) An algebraic number field is a field extension of Q of finite degree. Definition (group of automorphisms of L fixing K ) Aut( L / K ) = { σ ∈ Aut L : σ ( x ) = x for each x ∈ K } Definition (abelian extension) A number field extension L / K is said to be an abelian extension if: | Aut( L / K ) | = [ L : K ] i.e. a Galois extension Aut( L / K ) is commutative. Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 2 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 3 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Example √ Q ( − 5) ⊆ Q ( ζ 20 ) where ζ 20 = exp(2 π i · 1 / 20) . √ − 5 = 2 ζ 7 20 − ζ 5 20 + 2 ζ 3 20 . Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 3 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 4 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Class Field Theory Assume K = Q or K is a number field such that there does NOT exist an injective ring homomorphism σ : K ֒ → R . i.e. K has no real embeddings Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 4 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Class Field Theory Assume K = Q or K is a number field such that there does NOT exist an injective ring homomorphism σ : K ֒ → R . i.e. K has no real embeddings Class field theory tells us: The existence of a ray class field H K ( m ). Every finite degree abelian extension of K is contained in a ray class field H K ( m ). Information on the structure of Aut( H K ( m ) / K ). Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 4 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Class Field Theory Assume K = Q or K is a number field such that there does NOT exist an injective ring homomorphism σ : K ֒ → R . i.e. K has no real embeddings Class field theory tells us: The existence of a ray class field H K ( m ). Every finite degree abelian extension of K is contained in a ray class field H K ( m ). Information on the structure of Aut( H K ( m ) / K ). Aut( H Q ( m ) / Q ) ∼ = ( Z / m Z ) × H Q ( m ) = Q ( ζ m ) ζ m �→ ζ n m ↔ n ∈ ( Z / m Z ) × ζ m := exp(2 π i · 1 / m ) Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 4 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. H Q ( m ) = Q (exp(2 π i · 1 / m )) e = exp(2 π iz ) R S 1 ( C ) Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. H Q ( m ) = Q (exp(2 π i · 1 / m )) R / Z e = exp(2 π iz ) S 1 ( C ) Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. H Q ( m ) = Q (exp(2 π i · 1 / m )) R / Z e = exp(2 π iz ) S 1 ( C ) Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. H Q ( m ) = Q (exp(2 π i · 1 / m )) R / Z e = exp(2 π iz ) S 1 ( C ) Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. H Q ( m ) = Q (exp(2 π i · 1 / m )) R / Z e = exp(2 π iz ) S 1 ( C ) 0 1 2 3 4 5 6 6 6 6 6 6 Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14

Theorem (Kronecker-Weber Theorem) Every abelian extension L / Q with finite degree is contained in a field Q (exp(2 π iz )) for some z ∈ Q . Problem (Hilbert’s 12th Problem) Given a number field K, construct all finite abelian extensions of K by adjoining special values of particular analytic functions. H Q ( m ) = Q (exp(2 π i · 1 / m )) R / Z e = exp(2 π iz ) S 1 ( C ) Jared Asuncion Computing HCF’s of Quartic F’s using CM 28 November 2019 5 / 14