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College Algebra Exam Review 310

# College Algebra Exam Review 310 - 320 7 FIELD EXTENSIONS...

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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 F OExŁ has a complete set of roots in L (i.e., g.x/ factors into linear factors in LOExŁ ), 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 KOExŁ .
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