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m322fin - B U Department of Mathematics Spring 2009 Math...

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Unformatted text preview: B U Department of Mathematics Spring 2009 Math 322 - Algebra II, Final Exam, 24/5/2009, 11:30-13:00 Full Name Student ID : : over 40 A 1-1 ring homomorphism between two fields is called a field homomorphism. An onto field homomorphism is called a field isomorphism. A K-homomorphism between two extensions of a field K is by definition both a K -linear vector space homomorphism and a field homomorphism. The Hasse diagram of a group G is a graph in which vertices are subgroups of G and two vertices are connected by an edge if one is a subgroup of the other and if there is no other subgroup in between. 1. [10pts] (question by B.V.) In Z2 [x], find all irreducible polynomials of degree 7 over Z2 . 2. (a) [5pts] How many field isomorphisms are there from a field K to itself? (b) [5pts] Let E and F be two extensions over K . Show that a field homomorphism ϕ : E → K is a K -homomorphism if and only if ϕ fixes every element of K . √ (c) [5pts] How many Q-isomorphisms are there from the field Q( 3 2) to itself? (They are called Qautomorphisms.) 3. [2pts] Give two historical motivations to study field extensions. √ 4. Crash course on Galois theory. Consider K = Q(j, 3 2) where j = e2πi/3 . Remember that K is the splitting field of x3 − 2 over Q. (a) [2pts] Let n = |K : Q|. What is the value of n? Write down a basis {u1 , . . . , un } for K over Q. (b) [2pts]Plot the complex numbers u1 , . . . , un on the complex plane. (c) [5pts]Let σ1 , . . . , σk be the Q-automorphisms of K . Math Community knows that each σi takes • j to j or j 2 ; √ √ √ √ • and 3 2 to 3 2 or j 3 2 or j 2 3 2. What is the value of k ? Work out the relations between σi ’s to find the group with elements σi and with the composition operation? (d) [5pts] Draw the Hasse diagram of S3 . (e) [5pts] Draw the diagram of inclusions of all intermediate extensions between K and Q. 5. [0pt] Challenge. Can you prove (now or later) the fact in question (4c) above? ...
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