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Solvable Group - Mathematics - Exam, Exams of Mathematics

Acharya Nagarjuna UniversityMathematics

This is the Exam of Mathematics which includes Vector Valued Function, Denote, Statements, Vector Equation, Equation, Plane Containing, Derivatives of Inverse, Notation, Derivatives etc. Key important points are: Solvable Group, Serial Number, Subgroup, Separable Extension, Algebraic Closure, Nite Group, Unique Subgroups, Polynomial, Irreducible, Primes

Typology: Exams

2012/2013

Uploaded on 02/21/2013

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Math 501, Fall Term 2009
Final Exam
December
15
th
,
2009
Do not turn this page over until instructed
•
On the top of each exam booklet write
1. Your student ID;
2. The serial number of this exam;
3. The number of the booklet (if you use more than one).
•
On the sign-in sheet write
1. Your student ID;
2. The serial number of this exam;
3. Your name.
Instructions
•
You will have 150 minutes for this exam.
•
No books, notes or electronic devices.
•
Solutions should be written clearly, in complete English sentences, showing
all your work.
•
If you use a result from the lectures or the problem sets, quote it properly.
•
Good luck!
1
pf2

Partial preview of the text

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Math 501, Fall Term 2009

Final Exam

December 15 th, 2009

Do not turn this page over until instructed

  • On the top of each exam booklet write
    1. Your student ID;
    2. The serial number of this exam;
    3. The number of the booklet (if you use more than one).
  • On the sign-in sheet write
    1. Your student ID;
    2. The serial number of this exam;
    3. Your name.

Instructions

  • You will have 150 minutes for this exam.
  • No books, notes or electronic devices.
  • Solutions should be written clearly, in complete English sentences, showing all your work.
  • If you use a result from the lectures or the problem sets, quote it properly.
  • Good luck!

Math 501, Fall 2009 Final, page 2/

  1. (20 pts) Dene the following terms

(a) Sylow p-subgroup. (b) Solvable group. (c) Separable extension. (d) Algebraic closure.

  1. (15 pts) Let G be a nite group of order 561 = 3 · 11 · 17.

(a) Show that G has unique subgroups A, B of order 11 , 17 respectively. (b) Show that A and B are central in G. (c) Show that G is cyclic.

  1. (15 pts) Consider the polynomial t^3 + 2t + 1.

(a) Show that it is irreducible in F 3 [t]. (b) Show that it is irreducible in Q[t]. (c) Show that it is irreducible in F 32009 [t].

  1. (30 pts) Let p, q be primes and let f (x) = xp^ − q ∈ Z[x].

(a) Find a splitting eld Σ for f over Q. (b) Find [Σ : Q]. (c) Find Gal(Σ : Q).

  1. (10 pts) Let Q ⊂ K ⊂ L be number elds. Given α ∈ OL let g ∈ K[x] be the minimal polynomial of α over K. Show that g ∈ OK [x].
  2. (10 pts) Let r ∈ Q and let K = Q(cos(2πr)).

(a) Embed K in a radical extension of Q. (b) Show that K is normal over Q. (c) Show that Gal(K : Q) is Abelian. Hint: Express cos(2πr) using two roots of unity.