# HW 08 - MAT 322 Algebra with Galois Theory Professor Claus...

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Unformatted text preview: MAT 322: Algebra with Galois Theory Professor Claus Sorensen Problem Set 8 Rik Sengupta December 4, 2009 1. Let F be a field of characteristic p > 0, and fix an algebraic closure ¯ F . For each k > 0, show that the following subset is a subfield, F p- k = { x ∈ ¯ F : x p k ∈ F } ⊂ ¯ F. Let F per be the union of all these subfields. Show that F per is the smallest perfect subfield of ¯ F , containing F . That is, the perfect closure of F in ¯ F . Solution. Fix some k > 0. Then, suppose a,b ∈ F p- k . Then, from definition, a p k ∈ F , and b p k ∈ F . Then, we have, by the Frobenius mapping, ( a- b ) p k = a p k- b p k ∈ F , and ( a + b ) p k = a p k + b p k ∈ F . Also, obviously, ( ab ) p k = a p k · b p k ∈ F , and ( a- 1 ) p k = ( a p k )- 1 ∈ F . So, applying the definition once more, it clearly follows that a + b , a- b , ab and a- 1 are all in F p- k . Furthermore, 0 p k = 0, and 1 p k = 1, and so 0 and 1 are obviously in F p- k . So, it follows that F p- k is a field. Now, since for all a ∈ F p- k , we must have x p k- f = 0 for some f ∈ F , and so a is algebraic over F . So, F p- k is a subfield of ¯ F . 2 Now, note that F = F p- ⊆ F p- 1 ⊆ F p- 2 ⊆ .... Thus, F per is the union of an ascending chain of fields. Therefore, it is still a field, and in particular, we know F per is a subfield of ¯ F . Suppose we choose E to be the smallest subfield of ¯ F containing F that is perfect. In fact, since E is perfect, we know char E = char F = p , and so E p = E . Hence, we know F ⊆ E implies that all p th roots of the elements of F are in E . In other words, F p- 1 ⊆ E . By induction, then, F p- k ⊆ E for all k = 1 , 2 ,... . 1 So, it follows that F per ⊆ E . We will show now that F per is perfect. But this is immediate, because of the following reasoning. We know that char F per = p , which is obvious. Now, given any a ∈ F per , we must have a ∈ F p- k for some k , and then there exists some b ∈ ¯ F such that b p = a , and so, b p k +1 ∈ F , which implies that b ∈ F p- k- 1 ⊆ F per , i.e. for any a ∈ F per , there is a p th root of a also in F per . So, this is equivalent to proving that F per is perfect. But then, by the minimality of E , we must have E ⊆ F per . Hence, F per = E , i.e. F is the smallest perfect subfield of ¯ F containing F . 2 2. Let E/F be a finite extension, and assume E is a perfect field. Show that F is perfect. Is this true in general for algebraic extensions? Solution. Obviously, if char E = 0, then we must have char F = 0, and so F is perfect. So suppose char E = p > 0. Then, because E is perfect, we have E p = E . Also, char F = p > 0. Consider the ascending chain of fields F ⊆ F p- 1 ⊆ F p- 2 ⊆ ......
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HW 08 - MAT 322 Algebra with Galois Theory Professor Claus...

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