HOMEWORK 4
代做homework – 这个项目是homework代写的代写题目
All answers must be justified completely.
1. Let/be a quadratic extension of fields. (a) Ifchar() 2 , prove that=()for somesuch that^2 . (b) Ifchar()= 2 and/is separable, prove that=()for some such that ^2 +. 2. Prove that a fieldis perfect if and only if every finite extension ofis separable. (One direction was proved in class.) 3. Let= 21 /^2 + 21 /^4 R. Find the normal closure ofQ()/Q. 4. Let= 1 +^3
2 +^3
4. Find a normal closure ofQ()/Q.
5. Letbe a splitting field for^4 4 ^2 + 1 overQ. DetermineGal(/Q). 6. Letbe the splitting field of^6 + 1 overQ. DetermineGal(/Q). 7. Letbe the splitting field of^4 + 3 ^2 + 1 overQ. DetermineGal(/Q). 8. Find a subfieldof the field of rational functionsC()such thatC()/is Galois with Galois group isomorphic to 3. (Hint: Consider automorphisms ofC()of the form++with ,, , Z.) 9. Letbe a prime number. Assume that the degree of any finite field extension of a fieldis divisible by. Prove that the degree of any finite extension ofis a power of. 10. Letbe a field and letbe a prime number. Show that ifis not ath power, then is irreducible over. 11. Letbe an automorphism ofC(not necessarily of finite order), and letbe the fixed field ofC. Prove that every finite extension ofis cyclic. 12. Let/be a finite field extension and let. The trace and norm of, denotedTr/() andNm/(), respectively, are defined as follows. Let:be the map()= for . This a linear transformation of finite dimensional-vector spaces, so we can defineTr/()andNm/()to be the trace and determinant, respectively, of(in any -basis for). Now assume that/ is separable and let/be an extension such that there are =[:]field homomorphisms 1 , … ,:over. Prove that
Tr/()=
= 1
() and Nm/()=
= 1
().
Due : March 30 at 11:59pm.
2 homework 4
(Hint: First consider the case=()and relate everything to the minimal polynomial of
over. For the general case, consider the tower of extensions/()/.)