3207. FIELD EXTENSIONS – FIRST LOOKThese are generally known asCardano’s formulas, but Cardano creditsScipione del Ferro and N. Tartaglia and for their discovery (prior to 1535).What we will be concerned with here is the structure of thefield exten-sionKK.˛1; ˛2; ˛3/and the symmetry of the roots.Here is some general terminology and notation: IfFLare fields,we say thatFis asubfieldofLor thatLis afield extensionofF. IfFLis a field extension andSLis any subset, thenF.S/denotes thesmallest subfield ofLthat containsFandS. IfFLis a field extensionandg.x/2F OExŁhas a complete set of roots inL(i.e.,g.x/factors intolinear factors inLOExŁ), then the smallest subfield ofLcontainingFandthe roots ofg.x/inLis called asplitting fieldofg.x/overF.Returning to our more particular situation,K.˛1; ˛2; ˛3/is the split-ting field (inC) of the cubic polynomialf .x/2KOExŁ.
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