Algebraic number theory

Number fields and rings of integers: quadratic and cyclotomic fields, discriminants, ramification, ideals, adeles and ideles, class field theory.

• Quadratic fields and integer rings
• Cyclotomic fields
• Discriminants, ramification and ideals
• Adeles, ideles and class field theory

DefinitionShanks cubic polynomial#✓fully proved

For an element \(h\) of a ring \(R\), the Shanks cubic polynomial is

\[x^3 - h x^2 - (h + 3) x - 1 \in R[x].\]

noncomputable def shanksCubicPolynomial {R : Type*} [Ring R] (h : R) : Polynomial R :=
  Polynomial.X ^ 3 - Polynomial.C h * Polynomial.X ^ 2 -
    Polynomial.C (h + 3) * Polynomial.X - 1

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