{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

pma427_0708

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 3

(8 marks(v Explain why Q 1 1 Q = 3 6 or 9(3 marks(vi...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online