Unformatted text preview: roots of f.x/ . If ı 2 K , then dim K .E/ D 3 . Otherwise, dim K .E/ D 6 . Proof. Suppose that ı 2 K . Then Equation ( 7.4.4 ) shows that ˛ 2 and, therefore, also ˛ 3 is contained in K.˛ 1 / . Thus E D K.˛ 1 ;˛ 2 ;˛ 3 / D K.˛ 1 / , and E has dimension 3 over K . On the other hand, if ı 62 K , then we have seen that dim K .E/ is divisible by both 2 and 3, so dim K .E/ ³ 6 . Consider the ﬁeld extension K.ı/ ± E . If f.x/ were not irreducible in K.ı/ŒxŁ , then we would have dim K.ı/ .E/ ´ 2 , so dim K .E/ D dim K .K.ı// dim K.ı/ .E/ ´ 4; a contradiction. So f.x/ must remain irreducible in K.ı/ŒxŁ . But then it follows from the previous paragraph (replacing K by K.ı/ ) that dim K.ı/ .E/ D 3 . Therefore, dim K .E/ D 6 . n...
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- Fall '08
- Algebra, Quadratic equation, Elementary algebra, Prime number, Quintic equation