320 7. FIELD EXTENSIONS – FIRST LOOK These are generally known as Cardano’s formulas , but Cardano credits Scipione del Ferro and N. Tartaglia and for their discovery (prior to 1535). What we will be concerned with here is the structure of the ﬁeld exten-sion K ± K.˛ 1 ;˛ 2 ;˛ 3 / and the symmetry of the roots. Here is some general terminology and notation: If F ± L are ﬁelds, we say that F is a subﬁeld of L or that L is a ﬁeld extension of F . If F ± L is a ﬁeld extension and S ± L is any subset, then F.S/ denotes the smallest subﬁeld of L that contains F and S . If F ± L is a ﬁeld extension and g.x/ 2 FŒxŁ has a complete set of roots in L (i.e., g.x/ factors into linear factors in LŒxŁ ), then the smallest subﬁeld of L containing F and the roots of g.x/ in L is called a splitting ﬁeld of g.x/ over F . Returning to our more particular situation, K.˛ 1 ;˛ 2 ;˛ 3 / is the split-ting ﬁeld (in C ) of the cubic polynomial f.x/ 2 KŒxŁ . One noticeable feature of this situation is that the roots of
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.