Algebra - 2018

Please sign with your name and the date on which you had your exam. If you use this wiki, contribute to it as well or terrible things will happen to you: like me kicking you with my fists.

Andrin Hirschi, 9.8, 13:30-14:00

• Definition of integral domain
• Sketch of proof of existence of fraction field
• State the first isomorphism theorem for Rings and give an application

I showed (X,2)\modulo{Z}{4Z}[X] is a prime ideal

• State the correspondence between ideals of \modulo{R}{I} and ideals of R
• Definition of an UFD and state for an irreducible x the ideal xR is prime
• classify all maximal ideal of the real polynomials. Since R[X] is a PID the maximal ideals are exactly the ideals generated by irreducible polynomials. These are polynomials of degree 1 and polynomials of degree 2 without real roots.
• Mention an UFD which is not a PID. Just take K[X,Y] the polynomials in two variables over a field K.
• Is the Galoisgroup of the splitting field of a polynomial finite. Yes inject it into the symmetry group of it's roots
• State the 3 characterizations of a Galois extension.
• Prove one of them (2 implies 3 with notation of lecture notes)
• Prove that a Galois extension is simple. It suffices to show there are finitely many intermediary fields. The intermediary fields are bijection with the subgroups of the finite Galoisgroup.

Xavier, 06.08.2018, 08:00-08:30

Algebra I:

• Definition of ring, ring homomorphism. And describe homomorphism(s) \(\mathbb{Z}\to \mathbb{Z}\);
• Definition of integral domain, \(\mathbb{Z}/p\mathbb{Z}\) is such if and only \(p\) is prime (with proof)
• Polynomial ring and describe the homomorphisms \(R[X]\to S\)
• Definition of ideal, what is principal ideal and an example of a non-principal ideal
• Maximal ideal exists and give one on \(\mathbb{C}[X,Y]\)

Algebra II:

• Definition of the Galois group of a polynomial
• Definition of the splitting field: Why always exists and unique to isomorphism)
• Definition of polynomial having "no repeating root" (not asking separability)
• Galois group is trivial with non trivial extension ( \(X^p-t\) in \(\mathbb{F}_p(t)\))
• Galois group of \(X^3-2\)
• Definition of "solvable by radicals", definition of radical extension, definition of pure extension

I was the very first one I guess, so "lucky"...

Laurena Python, 06.08.2018, 09:30-10:00

Algebra I:

• Definition of integral domain, examples, Z/pZ is an integral domain if and only p is prime (with proof)
• What is a fraction field ? Sketch the construction
• Definition of prime and maximal ideal, examples, give a prime/maximal ideal of C[X,Y]
• State CRT and give a proof idea, definition coprime ideals
• Statement of 1st isomorphism theorem for rings

Algebra II:

• Definition of the Galois group of a polynomial, definition of the splitting field
• Definition of separablility (for a polynomial)
• What is the cardinality of the Galois group of a separable polynomial ?
• What can we say about the cardinality of the Galois group of an irreducible separable polynomial ? Why ?
• Definition of "solvable by radicals", definition of radical extension, definition of pure extension
• Galois group of X^3-2
• Galois group of F_p^n/F_p

They spend 20 minutes on Algebra I (Kowalski asks the questions) and 10 minutes on Algebra II (Burger asks the questions). They let you speak and do not interrupt, they also give useful hints. The proofs do not have to be written formally. If they see that you understand the concept, they sometimes stop you before the end.

Felix Sefzig, 06.08.2018, 11:00-11:30

Algebra I:

• Definition of integral domain with examples
• Construction of R[X] and describe the homomorphisms R[X]→S
• What is special about K[X]? Euclidean division and PID
• Find maximal & prime but not maximal ideals of C[X,Y,Z]
• What is conjugation & conjugacy classes
• Conjugacy classes in GL(n,C)
• Lagrange Theorem
• Definition of group action, G-morphism and examples
• 1st Isomorphism Theorem for groups
• Orbit-Stabilizer-Theorem

Algebra II:

• Definiton of Galois group, what can you say about the cardinality
• Galois group of F_{p^n}/F_{p}
• f irreducible, then f is separable iff gcd(f,f') = 1
• Normal extension, connection of Galois groups in tower of normal extensions
• Galois extension

Lukas, 06.08.2018, 13:30-14:00

Algebra I

• Groups: Examples and Homomorphisms between them
• GL(n,R) and the subgroups -> Special Linear Group SL(n,R) (and show that it's normal)
• for n=2; Non-normal subgroups of GL(n,R) (triangular matrices)
• Center and Conjugacy Class (and the connection Normal and Conjugacy class)
• Lagrange Theorem with proof, and application (Ord(x) | card(G))
• What is special subgroups of index 2? (they are normal, with proof)
• Does the converse of Lagrange hold too? (No, and no proof asked)

Algebra II

• Definition of Splitting field of Polynomial
• Gal(Fp^n : Fp) is isomorphic to Z/nZ, with proof
• Def of solvable by radicals

Manuel, 06.08.2018, 15:30-16:00

Algebra I

• Rings
  • Polynomial ring of a field: Polynomial division & Proof of why it works out
  • Proof that \( K[X] \) is a PID unsing the proven statment above
  • Definition UFD
  • Every PID is a UFD with Proof
• Groups
  • Definition of a group action with an example where \( G \) acts on itself
  • Orbit-stabilizer theorem + Start of proof (no time left)

Algebra II

• Definition Galois group of a Polynomial
• Definition Splitting field
• Definition Normal extension
• Wierd question about what i can say about the cardinalitiy of tha Galois group eventiully leading to...
• Given an irreducible polynomial of degree \( n \), i had to show that \( n\) divides the cardinality of the Galois group
• Define the three equivalences of a Galois extension (He wanted me to prove one direction, i think that the second one implies the third one but time didn't suffice)

All in all it seemed rather formal to me, K. as always is more straight forward and in my case more demanding, i guess, than B. which sometimes glances at K. with a smear grin when you say something funny in his opinion - made me is a bit unsetteled(perhaps just in my case bc he went for a cig break before). The exam flows exactly the way K. wants it in his part but it seems more free in B.'s Part.

Joël, 07.08.2018, 08:30-09:00

Algebra I:

• Show that \( \mathbb{Z}/m\mathbb{Z} \) is an integral domain iff \(p\) is prime. At one point I considered a prime decomposition \(m = p_1\ldots p_n\) and continued with the proof, whereupon he interrupted me and said that I must state \(n \geq 2\).
• Definition of the characteristic of a field, show that it must be prime.
• Definition of the Frobenius automorphism and show that it is a field morphism, ie show \(p \mid \)\({p}\choose{k} \) for \(k \neq 1, \; k \neq p\)
• Definition of polynomial rings as finite sequences in \(R\) with definition of addition, product, and which sequence corresponds to \(X\).
• Show that if \(R\) is an integral domain, then \(R[X]\) is an integral domain.

Algebra II:

• Definition of the Galois group of a polynomial, definition of the splitting field, why does it exist? Answer is Kronecker's theorem (without proof), which asserts that for every \(f \in k[X]\) there is a field \(L\) in which \(f\) splits, and then \(R(f) \subset L, \; k \subset k(R(f)) \subset L\) is a splitting field.
• Relation between the Galois group and the set of roots.
• The Galois group injects into \(S_{R(f)}\); cases where the Galois group is not isomorphic to \(S_{R(f)}\); Give an example of a polynomial whose Galois group is isomorphic to \(S_2\), I suggested \(X^2 + 1 \in \mathbb{Q}[X]\).
• Definition of a separable polynomial, \(|Gal(E/k)| = [E:k]\) in that case (without proof).
• Show that \(Gal(\mathbb{F}_{p^n}/\mathbb{F}_p) \cong \mathbb{Z}/n\mathbb{Z}\).
• Definition of a normal extension, characterisation of Galois extensions, without proof.

Chiara, 07.08.2018, 09:30-10:00

Algebra I:

• Def integral domain
• Z/pZ integral domain iff p prime with proof
• R integral domain -> R[X] integral domain with proof
• K field implies K[X] PID with proof
• Def polynomial ring
• Ideal in C[X] that is not prime
• Def algebraic extension, in connection with 1st isomorphism thm, def minimal polynomial, why it's irreducible
• Characteristic of a field with proof that p prime
• Chinese remainder thm with proof. Example where it's not surjective (look for ideals that are not coprime)

Algebra II:

• Def galois group & splitting field
• Galois group permutes the roots, injection into S_R(f)
• Example where it's surjective. I said Q with f in Q[X] irred, deg(f)=p, p-2 real roots. Did the proof. (Used p divides the cardinality of the Galois group, Cauchy, and that a transposition & a p-cycle generate Sp)
• Galois group of Fp^n/Fp with proof (did not have to do the whole proof)
• Burger asked what the dichotomy for X^p-c was, asked for the proof but 5 seconds after I started to think he stopped me because he thought it was too difficult. Connection where we used this in Lemma 3.9 in the proof, statement of Lemma 3.9 and application of Lemma 3.9

Laura, 7.08.2018, 11:00-11:30

Algebra 1

• Give 3 subgroups of Gl2(C)
• order of an element
• def group action
• 2 examples of group action where the group acts on itself
• def transitive action, give transitive action on GL2(C),determine #orbits
• finite groups: order of an element divides card of the group (with proof)
• how many groups exist of cardinality 17
• Lagrange (with proof), cosets
• Orbit stabilizer theorem
• def G-isomorphism
• First Iso Theorem for groups

Algebra 2:

• 3 equivalences for Galois extension
• Def fixed field
• Def separable polynomial, determine if the polynomial is separable
• proof Galois extension 1=>2 and 2=>3

Tomás, 07.08.2018, 14:30-15:00

Algebra I:

• Order of an element in a group.
• Lagrange Theorem (with proof).
• Prove order of element divides cardinality of finite group.
• 1st. Iso for groups with idea of proof.
• Definition of group actions.
• Can you have one group acting on a set in two different ways?
• Meaningful action of a group on subgroups: Kowalski wanted to hear 'conjugation', i couldn't come up with his answer, lost lots of time on it.
• OST with proof, time was up before we could finish it.

Algebra II:

• Definitions of Galois extension
• i) implies ii) (in the context of the three equivalent definitions). I couldn't prove it, Burger helped me all the way through.
• Galois group of \(X^3-2\). I proved it with Eisenstein and "p-2 roots", Burger was amused.
• Galois group of \(\mathbb{F}_{p^n}/\mathbb{F_p}\).

Patrik, 07.08.2018, 15:30-16:00

Algebra I:

• Defintion of a PID
• Give Examples of PID's
• Show that Z is a PID
• Maximal ideals in R[x]
• Definiton of a UFD
• Show that every PID is a UFD ( I started with the existence, but didn't have to finish it. He then wanted to hear how we prove existence. I said it is by induction and you need Euclid's Lemma. He asked me for Euclid's Lemma and didn't want to hear the rest of the proof)
• Chinese Remainder Theorem (Only idea of proof since there was no time left)

Algebra II:

• Definitions of Galois extension
• i) implies ii) (in the context of the three equivalent definitions).
• Defintion separable polynomial
• When does a polynomial have no multiple roots? (Without proof)
• Galois group of \(X^3-2\).

Emanuel, 07.08.2018, 17:00-17:30

Algebra 1

• Definition of PID
• Examples of PID (I gave \(\mathbb{Z}\) and \(K[X]\))
• Proof that one of my examples is actually a PID ( went with \(K[X]\))
• Example of Ring not PID with proof
• He then wanted to see what a UFD is, I gave him the definition. Is the decomposition unique? (no, up to units)
• What's the characteristic of a Field? What does it mean for a field to have char = 0? (he wanted to hear that it contains a copy of \(\mathbb{N}\), it told him it was infinite, he said he wanted a characterization)
• Can you name a infinite field of char = p? (I said the algebraic closure of \(\mathbb{F_p}\))
• Why is p prime or 0? (take kernel -> isomorphic to ID)
• State and prove CRT (he didn't want to see the part with the coprime intersection tho)
• What does it mean for an element to be algebraic?

Algebra 2

• Definition of Galois extension (I mentioned the 3 char, no proofs)
• Definition of seperable polynomial
• Solvability by radicals, give a definition for each concept used (solvable group etc.)
• He then asked me for the char. of solvable groups ( derived series )
• Definition of commutator subgroup, derived series ( I messed up here and went one step to far which cost a lot of time)

I messed up a lot when it came to excluding stuff like empty sets and special cases for zero and K. called me out on it every time. You have to be really careful with these things, it probably cost me a lot. In general be very precise with K. Having good, familiar examples on hand helps. Also Burger was really nervous for some reason and paced around in the room, so try not to get too distracted by that.

Hans, 07.08.2018, 13:30-14:00

Algebra 1

• xR = yR iff x = y*u for u a unit
• Chinese Remainder Theorem with proof
• K[X] a PID for K a field with proof
• Examples of PID
• Example of non-PID
• characteristic of a field with construction
• maximal Ideals in R[X]

Algebra 2

• Definition of Galois extension
• Definition solvable group + equivalent definition (derived series)
• Example of a polynomial with Galoisgroup S5 + proof
• Definition solvable by radicals

Berno Binkert, 08.08.2018, 09:30-10:00

Algebra 1

• Show that a ringhomomorphism between two fields is always injective.
• Definition of algebraic element.
• How does the kernel of \(ev_{\alpha}\) look?
• Does there exist an algebraic element of order 3 in \(\mathbb{R}\)?
• Definition of algebraic extension.
• Existence and uniqueness of finite fields.
• Why are finite fields closed under addition?

Algebra 2

• Definition of separable polynomial.
• Show for K a field with char(K) = 0, that every polynomial is separable.
• Definition of the Galois group.
• Is the Galois group finite?
• Give a Polynomial that has Galois group NOT isomorphic to \(S_{R(f)}\).
• I struggled with finding a polynomial. Since we didnt have much time, he then asked me what I know about \(Gal(F_{p^n}/F_p)\). He wanted me to realize that \(Gal(F_{p^n}/F_p)\) is NOT isomorphic to \(S_{R(f)}\).

Erik, 08.08.2018, 16:30-17:00

Algebra 1

• Construction of polynomial rings
• What can you say about homomorphisms from R[X] to S? (State the bijection \( Hom(R,S) \times S \simeq Hom(R[X],S) \) without proof)
• State the first isomorphism theorem (without proof)
• Is there a correlation between Ideals in R/I and ideals in R?
• Define PID, give examples and non-examples
• Prove that K[X] is a PID for any field K

Algebra 2

• Definition of the Galois group of a polynomial
• Why are Galois groups finite?
• State the three equivalent definitions of Galois extension
• Prove the implication (ii) \( \implies \) (iii)
• State the Galois correspondence theorem

Kowalski mechanically asks questions from a long list, stares impassively out of the window during your answer and still somehow notices every minor detail that you might have gotten wrong. Burger almost ran a marathon from one end of the room to the other while asking his questions. Both seemed to be really tired. The assistant Riccardo sits right next to you and when he sometimes nods a little, you know that you got your answer right.

Daisuke, 09.08.2018, 15:30-16:00

Algebra 1

• Example of an interesting non commutative ring? (symmetric group Sn)
• Example of infinite non commutative group? (Gln(C))
• Example of group Morphism from the group of units of R to (R, +), where R is the set of real numbers. I said exp from (R, +) to the group of units and it was ok.
• Lagrange Theorem with proof, application?
• Def Aut(G), Def Inn(G). Their relation?I said Inn(G) normal in Aut(G). Proof.
• First Isomorphism theorem for groups without proof.
• What can you say about the subgroups of G/H. I said that there is a bijection with the set of subgroups of G containing H. Proof of one direction.

Algebra 2

• Def Galois group of a polynomial
• Can the Galois group be infinite?
• Def separable polynomial
• Def Galois extension, proof of 2) => 3)

Helena, 09.08.2018, 16:00-16:30

Algebra 1

• Examples of groups
• Find a group homomorphism between (R,+) and (units of R,*), R the real numbers
• Def. group action, determine if a given action is transitive (for example, S_n acting on {1,...,n})
• Lagrange's Thm., with proof, show that left cosets are disjoint
• Orbit Stabilizer Thm., def. G-morphism
• Sylow Thm., without proof

Algebra 2

• Def. galois group of a polynomial
• Is the Galois group of a polynomial always finite?
• Galois extension: three equivalent definitions, proof of "2-->3"
• Galois correspondence Thm., proof of the 1st part
• Are Galois extensions always simple? Yes, ... .
• f solvable by radicals --> Gal(E/k) solvable, without proof

Benjamin Zayton, 15.08.2018, 09:00-09:30

Algebra 1

• Definiton of an integral domain, examples, non-examples
• Sketch of the construction of the fraction field of a domain
• Correspondence between ideals of R and ideals of R/I
• Definition of prime/maximal ideals and examples of prime/maximal/neither ideals in C[X,Y]
• Describe the structure of the quotient by x^2+1 in the setting of the previous exercise, this was my example for an ideal which isn't prime (I struggled with this one)
• Statement of the First Isomorphism Theorem
• Defintion of group actions, equivalent definition, examples
• Action of a group on its subgroups (Conjugation)
• Statement of the Orbit-Stabilizer Theorem, description of the G-isomorphism and of the action of G on the cosets of the stabilizer, definition of a G-isomorphism
• Application of the OST? (Sylow)
• Statement of the Sylow-Theorems, partial proof of the first statement

Algebra 2

• Definition of the Galois group of a polynomial
• Is it finite? (Yes, Injection to the symmetries of the roots via restriction)
• Example of a polynomial whose Galois group is not the whole symmetric group (x^(p^n))-x in F_p), proof of the example
• Definition of seperability
• Given an arbitrary seperable polynomial, show that it is seperable in an extension of the field (I gave the criterion for seperability of irreducibles with the gcd and produced a somewhat dubious idea for the proof)
• Characterization of Galois extensions, proof of 2) => 3)
• Statement of Hilbert 90, application?

Yaniv, 15.08.2018, 14:30-15:00

Algebra 1

• Definition of algebraic element (kernel of evaluation)
• existence and uniqueness of minimal polynomial. proof
• Definiton of algebraic extension
• show that every finite extension is algebraic.
• lagrange theorem with proof.
• show that intersection of cosets is empty
• name an application: order of element divides group. proof
• name a subgroup of \(S_n\) of index n (\(S_{n-1}\) viewed as subgroup of \(S_n\))
• state orbit stabilizer theorem and define G-morphism

Algebra 2

• Definition of the Galois group of a polynomial
• Is it finite? (Yes, Injection to the symmetries of the roots via restriction)
• Definition of separable
• is \(x^p+1\) separable over \(\mathbb{F}_p\)
• What can you say about \(Gal(\mathbb{F}_{p^n}/\mathbb{F}_p)\) (did the proof)
• 3 Characterization of Galois extensions, proof of 2) => 3)

Charlotte, 15.08.2018, 16:30-17:00

Algebra 1

• Definition of Aut(G) - why is it a group
• Definition of Int(G)
• Proof Int(G) is norml in Aut(G)
• State and proof Langrange theorem
• Classification of groups with Lagrange theorem (Groups with cardinality p are cyclic, proof)
• State Orbit-Stabilizer-Theorem, define G-morphism
• State Sylow-theorem, start proving the first part

Algebra 2

• Definition of the Galois group of a polynomial
• Is it finite? (Yes, Injection to the symmetries of the roots via restriction)
• f a seperable polynomial in k. Is it still seperabel in B for an extension B of k?
• 3 Characterization of Galois extensions, proof of 2) => 3)

Raphael, 16.08.2018, 11:30-12:00

Algebra 1

• Def alg element
• Why is f irreducible? (with proof)
• Def alg extension
• L finite ext, then alg (with proof)
• Subset of alg elements of algebraic extension is a field (with proof)
• K[X] is a PID (with proof)
• Def UFD
• Example of a non UFD
• PID => UFD (proof of existence)
• CRT (with proof)
• example of coprime ideals

Algebra 2

• Def Galois group
• Is it finite?
• Do you know a Galois group which isn't isomorphic to Sn? (Gal(Fpn/Fp) as always)
• Def separable polynomial?
• Is \(x^p+1\) separable in a field of char p?
• Galoisgroup which is isomorphic to Sn
• Generalization -> p prime, f irred of deg p , p-2 real roots... what did we use in the proof? I said Cauchy's Thm and stated it, then he asked me if I knew an easier proof than he did in class ... no I didn't.
• Galois extension and proof of 2=>3

Emie, 16.08.2018, 13:30-14:00

Algebra 1

• Lagrange with proof, proof cosets are disjoint
• example index n subgroup of Sn (Sn-1 in Sn)
• statement First and second isomorphism theorem of groups, give the map in the second
• connection of subgroups of G/H and subgroups of G, proof of one direction of the bijection (p^-1(p(K)) = K)
• defintion of group action, example (I gave GLn(Fp) on Fp^n, on itself with left multiplication/conjugation), give example when GxG acts on G
• Sn acts on {subsets of (1...n)} by sigma({i1,i2,i3...ik}) = {sigma(i1),sigma(i2)...sigma(ik)}. What are the orbits + proof
• statement orbit-stabilizer theorem, definition of G-morphism
• proof cardinality of Gln(Fp) as an application
• statement Sylow and proof of first part

Algebra 2

• definition Galois group of a polynomial, is it finite? (He asked if it works with the 0 polynomial as well)
• give 2 examples where the Galois group is whole Sn
• definition separable
• Is X^p+1 separable in Fp[X]
• three equivalent definitions of Galois extension
• Is a Galois extension simple?
• equivalence: simple/only finitely many intermediate extensions (he asked if I know an easy proof, I said no and gave the idea of simple -> only finitely many intermediate extensions)
• why is a finite field never algebraically closed? (I had no clue, Burger helped me through. The solution is: Take X^n-1, where n is bigger than the cardinality of the field and not divisible by p when char(k) = p)

Carlo, 16.08.2018, 14:30-15:00

Algebra 1

• Lagrange Thm with proof, proof that cosets are disjoint
• application of Lagrange: order of element divides order of group, and then characterization of groups of prime cardinality
• statement of 1st isom. Thm for groups and example of this. Here I messed something up, because I confused it with the same thm for rings, but it didn't seem to matter that much
• definition of group action, example (G acts on set of subgroups by conjugation)
• orbit stabilizer theorem and def. of G-morphism, how does G act on G/Stabilizer, and on T (where G acts on T)
• application of orbit stabilizer: Sylow Thm with proof of 1st part
• k field, why is k[x] a PID? How do you show the existence of the polynomial division?
• polynomial division of f in k[X] by (X-a), where a is an element of k

Algebra 2

• Definition of the Galois group of a polynomial
• Is it finite? (Yes, Injection to the symmetries of the roots via restriction)
• def. of f seperable, do we habe a criterion for when f is seperable? (yes, f is seperable iff (f,f')=1 always (I said if irreducible))
• Gal(Fpn/Fp), why is Xpn-X a seperable polynomial (just use criterion above)
• Why is a Galois simple with proof of one direction
• Galois correspondence (statement of first part)
• Statement of Hilbert 90, application (Galois ext of prime degree is pure of prime type)

K. and B. were laughing at something when I came in and seemed to be in good mood. They give you hints if needed, and if you say something wrong, they sometimes wait until you find out yourself. Sometimes they stop you while proving something if they see that you are on the right track

Skander, 16.08.2018, 17:00-17:30

I entered the room and both K and B were giving me a death glare showing me how tired they were of staying confined in this place repeating the same stuff over and over again, which led to an obviously very motivating start...

Algebra 1

• Definition of a Polynomial ring
• Write down the different operations
• show that the multiplication is associative (all of this Algebra for this ? Literally any other question would've been more relevant, I struggled and lost a lot of time on this as K did not budge )
• Question: "What else can you tell me about polynomial rings ?"
• I replied that I knew that if the coefficients are issued from a field it is a PID
• Proof that it is indeed a PID
• Proof of polynomial divisions with coefficients in a field (K was very picky there)
• CRT, show well-definedness (no comment)
• CRT, give proof idea which then turned into write down the entire Induction, I somehow managed to get it done before the time was up

Algebra 2

• What is the Galois group of a polynomial
• Define separability
• Give the criteria for separability (gcd with derivative, proof idea)
• Three equivalences of a Galois extension, proof of (1)=>(2)
• Write down the Galois correspondence

During K's interrogation, Burger was either spinning around in his chair while stretching his arms and legs or taking books from the shelf, not opening them and putting them back right away. During B's interrogation he let out a "yes I kno zis alredy, sank you" and I never figured out why. I honestly do not know how I should feel about this exam.