Extensions
Extensions of this Spiegelungssatz were given by Oriat and Oriat-Satge, where class groups were no longer associated with characters of the Galois group of K/k, but rather by ideals in a group ring over the Galois group of K/k. Leopoldt's Spiegelungssatz was generalized in a different direction by Kuroda, who extended it to a statement about ray class groups. This was further developed into the very general "T-S reflection theorem" of Georges Gras. Kenkichi Iwasawa also provided an Iwasawa-theoretic reflection theorem.
Read more about this topic: Reflection Theorem
Famous quotes containing the word extensions:
“The psychological umbilical cord is more difficult to cut than the real one. We experience our children as extensions of ourselves, and we feel as though their behavior is an expression of something within us...instead of an expression of something in them. We see in our children our own reflection, and when we don’t like what we see, we feel angry at the reflection.”
—Elaine Heffner (20th century)
“If we focus exclusively on teaching our children to read, write, spell, and count in their first years of life, we turn our homes into extensions of school and turn bringing up a child into an exercise in curriculum development. We should be parents first and teachers of academic skills second.”
—Neil Kurshan (20th century)