Algebra - 2019

Ole, 15.07.2019 13:30-14:00

Questions:

Algebra I

2. Let \( G = F^\times_{17} \) be the multiplicative group of the field \( F_{17} \) . Does there exist a transitive group action of G

(a) on a set with 8 elements?

(b) on a set with an odd number of elements?

7. Let G be a finite group. Let H be a subgroup of G which is not normal. Can the index of H be 2? Can the index of H be 3? \\

15. Does the alternating group \( A_{17}\) have any subgroup of index 2 or of index 3? Does the alternating group \( A_{17} \) have any elements of order 12?

29. Let \( \xi \) be the real third root of 5. Is the field \( Q(i, \sqrt{3}, \xi) \) the splitting field in \( C \) over \( Q \) of some polynomial \( f \in Q[x] \)? If so, give an example of such a polynomial.

39. Let \( p \) be a prime number. For every integer \( k \geq 1 \), define (see matrix from question sheet).

a) Show that \( A_k \) is a subgroup of GL\(_2(F_p) \)

b) Is \( A_k \) a normal subgroup of GL\(_2(F_p) \)?

c) What is the index of \(A_{p-1}\) in GL\(_2(F_p)\)?

44. Let \(a \in Z \setminus \{0, \pm 1\}\) be a square-free integer, that is, an integer which is not divisible by any perfect square except 1. Prove that for each \(n \in Z_{n>0}\), the polynomial \(x^n - a \in Q[x]\) is irreducible.

49. Let \(R=Z[x,y]\). Can you find a tower of ideals \(I_{0} \subset I_{1} \subset I_{2} \subset I_{3}\subset R\), where each of the \(I_{i}\) is a prime ideal?

50. Consider the cubic polynomial \(f(x)=(x^{2}+17x+1)(x-17)\in Z[x]\). For each prime p, let \(\bar{f}(x) \in F_{p}[x]\) be the polynomial defined by taking the coefficients of \(f\) mod p. For which primes does \(\bar{f}(x)\) have repeated roots in the algebraic closure of \(F_{p}?\)

Algebra II

3. Let \(G\) be a finite group. Does there always exist a finite dimensional extension of fields \(E/F\) which is Galois with Galois group \(G\)?

4. Let \(Q(\zeta)/Q\) be the extension of the rational numbers determined by \(\zeta = \exp(2\pi i/ 29) \in C\). Let \(\alpha \in Q(\zeta)\) be a number which is constructible over \(Q\). What are the possible values of \([Q(\alpha) : Q]\)?

20. Let \( f(x) = x^3 + 2x^2 + 3x + 4\). Let \(r_1 , r_2 , r_3 \in C\) be the three roots of \(f(x)\). Calculate \(r_1^3 + r_2^3 + r_3^3\)

The ambience is pretty laidback, you write on the blackboard while Prof. Pandharipande asks you questions and Ms. Jakob takes notes. He will often ask you follow-up questions to the question at hand and/or to your solution.