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 green grapes. I told him I prefer red ones and politely declined. 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 :)

Emanuele, 23.08.2019 11:30-12:00

Algebra I Questions: 15,20,29 and 46. He asked some follow up questions in particular regarding question 20. As I mentioned the existence of a primitive element for the extension \(\mathbb{F}_{2^n}/\mathbb{F}_2\), he asked me to prove it.

Algebra II Questions: 2,3 and 18. He didn't asked any particular follow up questions for this three.

The ambiance is a bit strange, Prof. Pandharipande stays at his desk and seems bored. Anyway there is no stress and you have all the time to present your solutions, sometimes Prof. Pandharipande moves on before you are done with the previous question.

Fabian, 23.08.2019 16:30-17:00

Algebra I

10d) What is the automorphism group of \( S_3\)?

I presented the solution where we only have to determine the images of the transpositions, and that every automorphism is a permutation of the three transpositions, hence the automorphism group is \(S_3\). He then asked, why only assigning transpositions will extend to a whole homomorphism. I struggled some time to explain to him, that what we are actually doing is just relabelling the elements "1","2","3", and that being a homomorphism really does not matter what names we give the elements. He finally accepted that explanation. "You were actually describing in a complicated way the action of \( S_3\) on itself by conjugation."

20a) Show that for every degree n, there exists an irreducible polynomial of degree n in \(\mathbb{F}_2[x]\).

I wanted to use that \(\mathbb{F}_{2^n}^\times\) is a cyclic group, then he asked why this is the case. So I began to prove Rotman Theorem A-3.59 with Theorem A-4.90, but in the middle he stopped me and went on.

14) Let G be the rotational symmetries of a regular solid cube. (a) How many elements does G have? (b) Does there exist a surjective homomorphism from G to \(S_3\)?

After I defined the homomorphism by the G-action on the three axes, he asked me what the kernel of this homomorphism was. I struggled to find out. So he asked how big the kernel is, and I tried something with the Orbit-Stabilizer-Theorem. "This does not help for the kernel! This question is actually not that hard!" So I realized, that the First Isomorphism Theorem gives us directly, that the kernel has order 4: "So it has to be an order 4 subgroup of \(S_4\)..." - "Well, both the Klein-Four-Group and \(\mathbb{Z}/4\mathbb{Z}\) are subgroups." I still couldn't find the solution, and Prof Pandharipande finally asked me: "Can't you actually see what the kernel is?" I didn't, and finally we went on. After the exam, I found out with a friend of mine that the kernel must be the Klein-Four-Group since it had to be a normal subgroup of \(S_4\).

29) Is the field \(\mathbb{Q}(i,\sqrt{3}, \sqrt[3]{5})\) a 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.

My solution was \(f=(x^2+1)(x^2-3)(x^3-5)\). He asked me what would happen if I left out \((x^2-3)\), and I found out that the splitting field still is \(\mathbb{Q}(i,\sqrt{3}, \sqrt[3]{5})\). "So you have been quite wasteful with your polynomial...".

Algebra II

5) Let E be the splitting field of the polynomial \(x^4− 2x^2− 3\) over \(\mathbb{Q}\). What is the degree of \(E/\mathbb{Q}\)? Find all proper subextensions of \(E/\mathbb{Q}\).

After I found the Galois group, I had to determine all subgroup of it, but I forgot the subgroup {(0,0), (1,1)}, so Prof Pandharipande gave me a hint, something like: "In mathematics, if you have two elephants, there is always the diagonal elephant". That hint actually helped XD.

And finally, the great finale:

25a) Let \(E/\mathbb{Q}\) be a finite dimensional field extension. 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\(F_1/\mathbb{Q} = 2\), dim\(F_2/\mathbb{Q} = 3\). Let \(z_1 \in F_1 \backslash \mathbb{Q}, z_2 \in F_2 \backslash \mathbb{Q} \). Must there exist \(\lambda_1, \lambda_2 \in \mathbb{Q}\) such that \( \mathbb{Q}(\lambda_1 z_1 + \lambda_2 z_2 = E\)?

After I gave the long proof of this, he asked me: "Did you come up with this solution?" - "I have to admit, it was a friend of mine who told me this solution" - "There is actually a much easier solution, where you don't explicitly construct the lambdas. There are only finitely many subextensions, and then you can argue that there must exist some lambdas..."

Conclusion: Hurry up with proving your results, don't waste too much time on proving or mentioning trivialities and just try to answer as many questions as you can. If you are too sketchy, Prof Pandharipande will probably stop you.

Tobias, 26.08.2019 08:30-09:00

Algebra I

10d) Did show that it was at least \(S_3\), but didn't know why there are not more Automorphims.

30) I had some serious problems with +,- which he founded funnier from time to time.

28) No questions

29) I gave the polynomial (again with wrong +,- for the roots, WTF was wrong with me). He wanted to know how I can find the other 3rd roots of 5. I said that we can construct the 3rd root of unity using root of 3 which he wanted to know exactly. I messed up with +,- again but found the correct result after some tries using Eulers Formula.

20) I argued that there are only finitly many intermediate extensions between \(F_2\) and \(F_{2^n}\) such that \(F_{2^n}\) is simple which he accepted. He wanted to know why there are only finitely many. I argued with the power set of \(F_{2^n}\). Then I took a generator of \(F_{2^n}\) and it's minimal polynomial.

Algebra II

5) Some really bad +,- errors agains, else no problems.

18)

Silvio, 26.08.2019 13:00-13:30

Algebra I:

43)

30)

40)

20)a) Here he wanted me to show any claim I made

14) "Do you know what \( G \) is? Explain why."

Algebra II:

2) "Show that the extension \(Q(\sqrt[3]{5},\zeta)/Q(\sqrt[3]{5}) \) has dimension \(\varphi(15)\)". Skipped b)

5)

25) For b: after I answered "no, because you can insert .." he interrupted me and said that the exam was over.

Joel, 26.08.2019 16-16:30

The assistent let me in 5 minutes early. Then we started directly:

Algebra I:

• 21) After I wrote down the first two, he asked how many 4-dim modules there are. (he was looking for "partition of 4"). I didn't get it and we moved on.
• 22) The only thing he asked during my proof is why \( e^{2 \pi i/3} \) is not in \( \mathcal{Q} (^3\sqrt{2}) \). I replied: Because it is a complex number. He was okay with that.
• 20) I wanted to start with b). He said: Forget about b). So I started to prove a) with the help of the "principle element theorem". After a while, he asked me to stop, saying "you understood the exercise".

Algebra II:

• 2) He wanted a good argument on why the degree has to be 40. (He wanted to hear: it has to be at least 40 because it is the lcm(5,8) but also it has to be less or equal to 40, since if one of the polys would factor, the degree can only be reduced.)
• 25) After I solved a) using again the "principle element theorem", he asked about b). After a short explanation, he was satisfied with my answer and let me leave.

The exam only took about 20 minutes. The atmosphere is very quiet. He sits behind his desk and rarely interrupts.

Viola, 27.08.2019 11:30-12:00

Algebra I:

30) He wanted to know exactly why the splitting field is \(\mathbb{Q}(\sqrt{2},\sqrt{3})\)

10 b) I determined the automorphism group then he wanted me to identify it, it is the Klein 4 Group.

21) I did a) and b) and then he wanted to know how many modules there were of dimension 3 and 4, I explained to him that the structure always is a partition of the dimension.

29) I directly stated the polynomial \((x^2 +1) (x^3-5)\) and then explained why the third root of unity is in the splitting field. He was happy with that and we moved on to Algebra II.

Algebra II

1) I argued that the cyclotomic polynomial has degree 6 and is irreducible over \(\mathbb{Q(\alpha)}\), he told me we had only shown in class that it is irreducible over \(\mathbb{Q}\). I Said “in that case you would probably need to show that it is greater / equal than 6 and then show it is less / equal to 6” at which point he moved on to the next question.

5) He had no further questions.

3) He asked me if I knew the proof of why the Galois group of \(\mathbb{C}(x_1, ..., x_n) / \mathbb{C}(e_1,...,e_n)\) = \(S_n \) .. I didn’t know it but reasoned a bit why it makes sense by pointing out the form of the symmetric functions and that the Galois group would need to fix those

He then said the exam was over and let me go 8 minutes early.

Nico, 28.08.2019 10:30-11:00

Algebra I:

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.

First I stated the generell form and then deduces from this to a). As I started part b), he asked, if there is an isomorphism between the two options I wrote at the blackboard. I didn't give an answer so he let me go to the next question after a couple of hints which I didn't use. We skipped part c).

43) Is there a transitive action of \( S_5 \) on a set with 10 elements? Is there a transitive action of \( S_5 \) on a set with 20 elements?

Here I gave the cosets of \( A_4 \) and \( S_3 \) to have a set with the required number of elements. He asked why these actions are transitive. As I said, that in a group we can reach every element from another element by multiplying it with the right, and so we get in all cosets, so he was happy.

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

I presented the way of how we can solve this question. Then I just needed to show the case \( x^2 + 17x +1 = (x - \alpha )^2 \). As I finished the exercise, he was surprised, that I knew the square of 17 by heart.

Algebra II

2) Let \( K = \mathbb{Q}(\sqrt[3]{5}, \zeta) \), where \( \zeta \) is a primitive \( 15^{th} \) root of unity. (a) Prove that \( K \) is Galois over \( \mathbb{Q} \) and find the order the Galois group \( G \). (b) Show that \( Gal(K/ \mathbb{Q}(\zeta)) \) is a normal subgroup of \( G \).

I forgot to show first, that we have a Galois extension, which we need later in the exercise to claim. So he let me do my stuff and then interrupted me and asked why we can do the extensions I presented. He helped me to find out, that K is Galois and so we could conclude. We skipped part b).

8) Is the regular 272-gon constructible by straightedge and compass?

I made some small mistakes when I wrote the formula for constructible polygons at the blackboard. He again helped me to find my errors and then I found the result he wanted me to find.

The atmosphere was really relaxed. First he asked me if I am another student as I am, so I needed to correct him before we started. To work at the blackboard during the exam is kind of fun, since I didn't do this a lot. He said the exercise m random picking in his head, so he said sometimes a number and then corrected himself again and said, he mentioned another question. At the end he asked which part I liked more of the lecture. I said Algebra II, since it is more beautiful for me than the Algebra I.

Jingi, 28.08.2019 17:00-17:30

Algebra I:

21)

14) + find the kernel of b)

30)

40)

Algebra II

8) construct 17-gon (his hint: observe Q(17-th root of unity)

1)

After these questions there are still 10 minutes left. Rahul asked me if I was able to do all the questions on the list. I said yes and he let me go after doing some small talk over rep theory (MMP2).

Kirill, 29.08.2019 14:00-14:30

First I head a really nice chit-chat with the assistant, while Professor was gone for a couple of minutes. Which really helped me to calm down;) The overall ambiance was really relaxing, however, I felt, like Prof wanted to speed up me (probably due to time restrictions). But it was fine. I had Algebra 1:

30)

14) he asked about the kernel of the homo and asked to show on imagined cube)

21) he asked whether C/(x^2) and C/(x)*C/(x) are isomorphic as C-modules. (No, take f(x)=x*f(1)=0 kernel not trivial)

Algebra 2:

3) he asked whether I can prove that Galois group of C(x1,..,xn)/C(e1,...,en) is S_n. I stated two main ideas of the proof and was happy. He also asked me to sketch the proof of 13 in algebra 1 (since I used it)

25)