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.

Viera, 19.08.2019, 13:30-14:00

Algebra I:

6. Let d be the smallest index of a proper subgroup of \(S_8\). What is d? Do there exist two distinct subgroups of \(S_8\) of index d?

9. Which of the following rings are fields? a) \(\mathbb{Z}[x]/(3,x^2+x+1)\) b) \(\mathbb{Z}[x]/(4,x^2+x+1)\) c) \(\mathbb{Z}[x]/(2,x^2+x+1)\) Explain why or why not.

19. Compute the conjugate of the group element a=(3 5 2 4)\(\in S_5\) by the element (2 3). How many elements of \(S_5\) are conjugate to a?

21. Let A be a module over the ring \(\mathbb{C}[x]\). Since \(\mathbb{C}\subset \mathbb{C}[x]\), A is a \(\mathbb{C}\)-vector space.

a) List all \(\mathbb{C}[x]\)-modules A with dim\(_\mathbb{C}A=1\), up to isomorphism.

b) List all \(\mathbb{C}[x]\)-modules A with dim\(_\mathbb{C}A=2\), up to isomorphism.

c) List all \(\mathbb{C}[x]\)-modules A with dim\(_\mathbb{C}A=3\), up to isomorphism.

29. Let \(\zeta=1.7099759466...\) be te real cube root of 5. Is the field \(\mathbb{Q}(i,\sqrt{3},\zeta)\subseteq \mathbb{C}\) the splitting field in \(\mathbb{C}\) over \(\mathbb{Q}\) of some polynomial \(f\in\mathbb{Q}[x]\)? If so, give an example of such a polynomial \(f\).

40. Let G be a group of order \(3^{17}\), and let \(H\subset G\) be a subgroup of index 3. Must H be normal?

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

Algebra II:

4. Let \(\mathbb{Q}(\zeta)/\mathbb{Q}\) be the extension \(\mathbb{Q}(\zeta)\subset\mathbb{C}\) of the rational numbers determinded by \(\zeta=\exp(2\pi i/29)\). Let \(\alpha\in\mathbb{Q}(\zeta)\) be a number which is contructible over \(\mathbb{Q}\). What are the possible values of dim\(_\mathbb{Q}\mathbb{Q}(\alpha)/\mathbb{Q}\)?

9. Let \(\mathbb{C}(e_1,e_2,e_3)\subset\mathbb{C}(x_1,x_2,x_3)\) be te subfield of symmetric functions where \(e_1=x_1+x_2+x_3, e_2=x_1x_2+x_1x_3+x_2x_3, e_3=x_1x_2x_3\). What is the degree of the extension of \(\mathbb{C}(e_1,e_2,e_3)\) generated by the element \(x_1+2x_2+3x_3\in\mathbb{C}(x_1,x_2,x_3)\)?

25. Let \(E/\mathbb{Q}\) be a finite-dimensional field extension of \(\mathbb{Q}\).

a) Suppose dim\(E/\mathbb{Q}=6\) and \(\mathbb{Q}\subset F_1\subset E, \mathbb{Q}\subset F_2\subset E\) are subextensions with dim\(F_1/E=2\), dim\(F_2/E\)=3. Let \(z_1\in F_1\) and \(z_2\in F_2\) with \(z_1,z_2\not \in \mathbb{Q}\). Must there exists \(\lambda_1,\lambda_2\in\mathbb{Q}\) such that \(\mathbb{Q}(\lambda_1 z_1+\lambda_2z_2)=E\)?

b) Suppose dim\(E/\mathbb{Q}=12\) and \(\mathbb{Q}\subset F_1\subset E, \mathbb{Q}\subset F_2\subset E\) are subextensions with dim\(F_1/E=4\), dim\(F_2/E\)=3. Let \(z_1\in F_1\) and \(z_2\in F_2\) with \(z_1,z_2\not \in \mathbb{Q}\). Must there exists \(\lambda_1,\lambda_2\in\mathbb{Q}\) such that \(\mathbb{Q}(\lambda_1 z_1+\lambda_2z_2)=E\)?

At the beginning he chooses the questions randomly, towards the end he asks you specific questions. He didn't really ask any follow-up questions, though he did ask about some points in my proofs. When he sees that you understand the proof, he sometimes just stops you and asks the next question.