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(8 marks(v explain why q 1 1 q = 3 6 or 9(3 marks(vi

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Unformatted text preview: (8 marks) (v) Explain why [ Q ( 1 ; 1 ) : Q ] = 3 ; 6 or 9 . (3 marks) (vi) Assuming that 1 62 Q ( 1 ) , prove that [ Q ( 1 ; 1 ) : Q ] = 9 . (3 marks) 2 (i) Let f ( x ) 2 Q ( p 2)[ x ] denote the quartic f ( x ) = x 4 + p 2 x 5 = 4 : In this particular numerical example, write down an algorithm to nd all the roots of f ( x ) . In particular, your algorithm should determine explicitly the resolvent cubic of this f ( x ) . (Hint: The resolvent cubic of f ( x ) lies in Z [ x ] : ) (15 marks) (ii) Let f ( x ) be as in (i). Find all the roots of the resolvent cubic of f ( x ) . (10 marks) PMA427 1 Turn Over PMA427 3 Let L=K be a nite eld extension. (i) De ne the Galois group Gal( L=K ) . (3 marks) (ii) Prove that j Gal( L=K ) j 6 [ L : K ] . (8 marks) (iii) What is meant by saying that L=K is a Galois extension? (2 marks) (iv) Give an example of a nite extension L=K which is Galois and one which is not. (4 marks) (v) State, without proof, the Fundamental Theorem of Galois Theory. (3 marks) (vi) The dihedral group of order eight D 8 has two generators r and s which satisfy the relations r 4 = s 2 = 1 ; and r 3 = srs: You may assume that there are precisely 10 subgroups of D 8 given by f 1 ; rs; r 2 ; r 3 s g ; f 1 ; r 3 s g ; f 1 ; rs g ; f 1 ; s g ; f 1 ; r 2 s gf 1 ; s; r 2 ; r 2 s g ; D 8 ; f 1 ; r; r 2 ; r 3 g ; f 1 ; r 2 g ; f 1 g : Suppose that L=K is a Galois extension with Gal( L=K ) = D 8 . Prove that there are precisely six elds M such that...
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(8 marks(v Explain why Q 1 1 Q = 3 6 or 9(3 marks(vi...

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