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.

Wayne, 22.08.2019, 11:00-11:30

You give your legi to Tim, who kindly offers you grapes. You present your solutions on the blackboard.

Algebra I:

20. a) Let \(\mathbb{F}_2\) be the field with 2 elements. Show that for every degree \(n \geq 1\) that there exists an irred. poly. of degree \(n\) in \(\mathbb{F}_2[x]\).

We spent ages on this one and didn't have time for the second part.

32. Let R be an integral domain with finitely many elements. Must R be a field? Answer with proof.

39. check the questions list bc latex doesn't want to work for matrices.

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

I gave the idea of the proof for this and he let me move on to algebra ii.

Algebra II:

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

(Tip: Please do not confuse the definition of solvable with constructible. It doesn't look good. He skipped the question afterwards after looking at me in a rather patronising way.)

12. Factorise \(x^8 - 1\) and \(x^9 - 1\) into irred. factors in \(\mathbb{Q}[x]\)

Here he made me explicitly draw out the unit circle and label which ones roots were part of which irreducible factor, which was a bit strange.

At the end of the exam, Panda asked me if I had enjoyed the course, to which I replied "yeah it was better than Measure and Integration", to which he laughed. He then asked me if I was a mathematics student, which was definitely the highlight of the semester. Be confident and take your time in answering the questions. There are a few follow up questions but once he sees you understand the question, he will let you move on.

Enjoy the holidays everyone :)