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Botham is among the 36 new peers, who also include former chancellors Ken Clarke and Philip Hammond, announced on Friday.
All-rounder Stokes' unbeaten 135 at Headingley helped the hosts chase down a record target of 359 in dramatic fashion and level the five-match series at 1-1.
An incredible display from Stokes, in an innings that included eight sixes, saw England snatch victory from Australia, levelling the series at 1-1 after their highest-ever successful run chase in any Test.