English
For K1, K2, K3 with normality, applying the restrictNormalHom to a base f is equal to composing the restrictions and then applying.
Русский
При нормальности K1, K2, K3 применение restrictNormalHom к базовому f равно композиции ограничений и последующему применению.
LaTeX
$$$\text{restrictNormalHom}_{K_1}(f) = (\text{restrictNormalHom}_{K_1})\circ (\text{restrictNormalHom}_{K_2} f)$$$
Lean4
/-- Normalized trace defined purely in terms of the degree and the next coefficient of the minimal
polynomial. Could be an alternative definition but it is harder to work with linearity. -/
theorem normalizedTrace_minpoly (a : K) :
normalizedTrace F K a = ((minpoly F a).natDegree : F)⁻¹ • -(minpoly F a).nextCoeff :=
have ha : IsIntegral F a := IsIntegral.isIntegral a
IntermediateField.adjoin.finrank ha ▸ trace_adjoinSimpleGen ha ▸ normalizedTrace_def F K a
