322final - F 5 . (10 points) 7 8. Let E/K be a feld...

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MATH 322 Final Examination Name: June 2, 2001 Surname: 9:00–12:00 Signature: TB 120 Question Score 5 /5 points 10 /20 points 1 /10 points 6 /5 points Bonus Question /20 points 2 /15 points 7 /10 points 3 /5 points 8 /15 points 4 /5 points 9 /10 points Total /100 points 1. Express the symmetric polynomial x 3 + y 3 + z 3 + xyz over Z in terms of the elementary symmetric polynomials. (10 points) 1
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2. Let A := ½µ a b c d Mat 2 × 2 ( C ) : µ a b c d ¶µ 0 i 1 1 = µ 0 i 1 1 ¶µ a b c d ¶¾ Mat 2 × 2 ( C ) . Is A a vector space over R ? Over C ? (15 points) 2
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3. Let K be a feld. Give the defnition oF the characteristic of K . (5 points) 3
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4. Let K be a feld oF characteristic p 6 = 0. Prove that ϕ : K -→ K , a 7-→ a p is a feld homomorphism. (5 points) 4
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5. Find the multiplicative inverse of 2 + 3 5 in the ±eld F 7 ( 5). (5 points) 5
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6. Let E/K be a feld extension and let D be an integral domain such that K D E . Prove that, iF E is algebraic over K , then D is a feld. (5 points) 6
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7. Find the number of monic irreducible polynomials of degree 72 over
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Unformatted text preview: F 5 . (10 points) 7 8. Let E/K be a feld extension and let L be an intermediate feld. IF L is ( K, E )-stable, prove that L 00 is ( K, E )-stable, too. (15 points) 8 9. Find an algebraic feld extension that is not separable. (10 points) 9 10. Let ζ := e 2 πi 12 ∈ C and E := Q ( ζ ). Find all subgroups of G := Aut Q E , all intermediate ±elds of E/ Q , and a primitive element for each of the intermediate ±elds. Describe the Galois correspondence by Hasse diagrams of groups and inter-mediate ±elds. (20 points) 10 Bonus question. Tell about the contribution to algebra of one of the following mathematicians: Gauss, Cauchy, Abel, Liouville, Klein, Kronecker, Dedekind, Heinrich Weber, Artin. 11...
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322final - F 5 . (10 points) 7 8. Let E/K be a feld...

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