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Such that k m l with m=k galois(5 marks 4(i what does

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Unformatted text preview: such that K M L with M=K Galois. (5 marks) 4 (i) What does it mean to say that a nite group G is soluble? (4 marks) (ii) Let K be a eld and f ( x ) 2 K [ x ] . What does it mean to say that the equation f ( x ) = 0 is soluble by radicals over K ? (4 marks) (iii) Let f ( x ) 2 Q [ x ] be an irreducible, monic polynomial of degree ve, having ve distinct roots three being real roots and two not. Let L be the splitting eld of f ( x ) over Q . Prove that Gal( L= Q ) is isomorphic to S 5 , the symmetric group of permutations of the set f 1 ; 2 ; 3 ; 4 ; 5 g . (You may assume group-theoretic properties of S 5 without proof.) (9 marks) (iv) Prove that the polynomial f ( x ) = x 5 10 x + 2 satis es the conditions of part (iii). (8 marks) PMA427 2 Continued PMA427 5 (i) Let K be a eld and f ( x ) 2 K [ x ] . Let L=K be a eld extension. What is meant by saying that L is a splitting eld for f ( x ) over K ? (3 marks) (ii) Let L=K be a nite eld extension which is Galois with Galois group Gal( L=K ) . Prove that L is a splitting eld for some polynomial f ( x ) 2 K [ x ] , stating carefully any results which you use without proof. (10 marks) (iii) Let p be a prime with positive square root p p . Let ! = 1 + p 3 2 and = p p + ! . Show that Q ( p p; ! ) = Q ( ) . (5 marks) (iv) Assuming that p p = 2 Q ( ! ) , nd the minimal polynomial of over Q . (You do not need to simplify it algebraically.) (4 marks) (v) Explain why your answer to part (iv) gives a monic, irreducible polynomial in Z [ x ] which does not satisfy Eisenstein's Irreducibility Criterion for any prime q . (3 marks) End of Question Paper PMA427 3...
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