ich ▁was ▁the ▁first ▁to ▁give ▁a ▁completely ▁self - cont ained ▁formula ▁for ▁the ▁pair ing ▁which ▁coinc ides ▁with ▁the ▁wild ▁Hil bert ▁symbol ▁on ▁local ▁fields , ▁thus ▁initi ating ▁an ▁important ▁branch ▁of ▁the ▁study ▁of ▁explicit ▁formulas ▁in ▁number ▁theory . ▁Another ▁famous ▁result ▁is ▁Sh af are v ich ' s ▁theorem ▁on ▁sol v able ▁Gal ois ▁groups ▁giving ▁the ▁real ization ▁of ▁every ▁finite ▁sol v able ▁group ▁as ▁a ▁Gal ois ▁group ▁over ▁the ▁rational s . ▁Another ▁fundamental ▁result ▁is ▁the ▁Gol od – Sh af are v ich ▁theorem ▁on ▁to wers ▁of ▁un ram ified ▁extensions ▁of ▁number ▁fields . ▁ ▁Sh af are v ich ▁and ▁his ▁school ▁greatly ▁contributed ▁to ▁the ▁study ▁of ▁algebraic ▁geometry ▁of ▁surfaces . ▁He ▁initi ated ▁a ▁Moscow ▁sem inar ▁on ▁classification ▁of ▁algebraic ▁surfaces ▁that ▁updated ▁around ▁ 1 9 6 0 ▁the ▁treatment ▁of ▁bir ational ▁geometry , ▁and ▁was ▁largely ▁responsible ▁for ▁the ▁early ▁introduction ▁of ▁the ▁scheme ▁theory ▁approach ▁to ▁algebraic ▁geometry ▁in ▁the ▁Soviet ▁school . ▁His ▁investigation ▁in ▁arithmetic ▁of ▁elli ptic ▁curves ▁led ▁him ▁independently ▁of ▁John ▁T ate ▁to ▁the ▁introduction ▁of ▁the ▁most ▁myster ious ▁group ▁related ▁to ▁elli ptic ▁curves ▁over ▁number ▁fields , ▁the ▁T ate – Sh af are v ich ▁group ▁( us ually ▁called ▁' S ha ', ▁written ▁' Ш ', ▁his ▁C yr ill ic ▁initial ). ▁He ▁introduced ▁the ▁Gro th end ieck – O gg –