{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

College Algebra Exam Review 423

# College Algebra Exam Review 423 - 9.4.3 By the induction...

This preview shows page 1. Sign up to view the full content.

9.4. SPLITTING FIELDS AND AUTOMORPHISMS 433 Corollary 9.4.15. If K L is a finite–dimensional Galois extension and K M L is an intermediate field, then M L is a Galois extension. Proof. L is the splitting field of a separable polynomial over K , and, there- fore, also over M . n Proposition 9.4.16. If K L is a finite–dimensional Galois extension, then dim K L D j Aut K .L/ j : (9.4.3) Proof. The result is evident if K D L . Assume inductively that if K M L is an intermediate field and dim M L < dim K L , then dim M L D j Aut M .L/ j . Let ˛ 2 L n K and let p.x/ 2 KOExŁ be the minimal polynomial of ˛ over K . Since L is Galois over K , p is separable and splits over L , by Theorem 9.4.14 . If ' 2 Iso K .K.˛/; L/ , then '.˛/ is a root of p , and ' is determined by '.˛/ . Therefore, deg .p/ D j Iso K .K.˛/; L/ j D OE Aut K .L/ W Aut K.˛/ .L/Ł; (9.4.4) where the last equality comes from Equation (
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: 9.4.3 ). By the induction hy-pothesis applied to K.˛/ , j Aut K.˛/ .L/ j D dim K.˛/ L is ﬁnite. Therefore, Aut K .L/ is also ﬁnite, and j Aut K .L/ j D deg .p/ j Aut K.˛/ .L/ j D dim K .K.˛// dim K.˛/ .L/ D dim K L; where the ﬁrst equality comes from Equation ( 9.4.4 ), the second from the induction hypothesis and the irreducibility of p , and the ﬁnal equality from the multiplicativity of dimensions, Proposition 7.3.1 . n Corollary 9.4.17. Let K ± L be a ﬁnite–dimensional Galois extension and M an intermediate ﬁeld. Then j Iso K .M;L/ j D dim K M: (9.4.5) Proof. j Iso K .M;L/ j D Œ Aut K .L/ W Aut M .L/Ł D dim K .L/ dim M .L/ D dim K M;...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online