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

This preview shows pages 1–11. Sign up to view the full content.

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

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

View Full Document
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
3. Let K be a feld. Give the defnition oF the characteristic of K . (5 points) 3

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

View Full Document
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
5. Find the multiplicative inverse of 2 + 3 5 in the ±eld F 7 ( 5). (5 points) 5

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

View Full Document
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
7. Find the number of monic irreducible polynomials of degree 72 over

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

View Full Document

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: 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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 11

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

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online