Algebra 2 - 2014/15

Past papers

Aside from some exceptions mentioned below, all questions on past exams should in principal be do-able, though there may be a few differences in notation which could cause confusion. There are probably a number of questions in the past papers which you could understand the solution to, but couldn't produce yourself in the exam because of the different techniques emphasised in this course.

If other problematic questions are identified (in tutorials, on duo, et cetera), the page will be updated.

2014, 2013, 2012, 2011, 2010, 2009


2014

Notation: \( \overline{a} \) is another way of writing cosets in a quotient ring (or group); the explicit notation can be worked out from context of the question. For example in 1) we have \( g(x) = \overline{2}x^4 + \overline{2}x^3 + \overline{1} \in \mathbb{Z}_5[x] \), so \( \overline{2} \) must mean \( [2]_5 \) since it lives in the coefficient ring \( \mathbb{Z}_5 \).

Secion A

1 - Should be able to do this, but Dr Häsä has explicily said on duo that he would not ask this type of question.

3 - Safe to ignore. This questions requires the full abstract Chinese remainder theorem, which has not been covered this year. If you would like to try anyway, the theorem says:

Let \( R \) be a commutative ring. Let \( I_1 \) and \( I_2 \) be ideals such that the ideal sum \( I_1 + I_2 = R \) (or equivalently \( 1 = i + j \) for some \( i \in I_1 \) and \( j \in I_2 \)). Then we call \( I_1 \) and \( I_2 \) coprime (think about \( R = \mathbb{Z} \) to see why), and have the following isomorphism of rings \[ R/(I_1 I_2) \cong R/I_1 \times R/I_2 \, , \] where \( I_1 I_2 \) is the ideal generated by all products \( a b \), with \( a \in I_1 \) and \( b \in I_2 \). (Products of generators is sufficient)

6ii - This requires two theorems which have not been presented in the lecture notes this year. The first theorem needed is the statement of Sheet 19 Challenging Question 10 about groups of order \( 2p \), for \( p \) prime. The second theorem deals with groups of order \( p^2 \) for \( p \) prime:

Let \( p \) be prime. Then any group of order \( p^2 \) is abelian, so is in particular isomorphic to either \( \mathbb{Z}_{p^2} \), or \( \mathbb{Z}_p \times \mathbb{Z}_p \).
Using conjugacy classes, show the centre cannot have order \( 1 \) or order \( p \). (Using the class equation of a group, which is just breaking the group into counjugacy classes and counting their sizes, you can exclude \( 1 \) easily. Excluding \( p \) is tricker, you need to know some facts about \( Z(G) \). For example, show that if \( G/Z(G) \) is cyclic, then \( G \) is abelian.)

Section B

7 - Dr Häsä requries all rings to have a (multiplicative) identity, so an ideal \( I \) does not actually form a ring. Therefore it doesn't make sense to speak of the polynomial ring with coefficients in \( I \) because \( I[X] \) does not have an identity and so is not a ring. If you attempt this question, drop the requirement that multiplication has an identity from the definition of a ring.

8 - Since \( \phi \) maps to \( \mathbb{Z}_2 / \langle f(x) \rangle \), we must have that \( \overline{g(x)\cdot g(x)} \) means the coset \( g(x)\cdot g(x) + \langle f(x) \rangle \)


2013

Notation: \( \mathbb{Z}/n \) is just another way to write \( \mathbb{Z}_n = \mathbb{Z}/n\mathbb{Z} \).

Notation: We would normally write \( \langle a_1, \ldots, a_n \rangle \), with angle brackets, for the ideal generated by \( a_1, \ldots, a_n \). Here it is written \( (a_1, \ldots, a_n) \), with round brackets.

Notation: \( \overline{a} \) is another way of writing cosets in a quotient ring (or group); the explicit notation can be worked out from context of the question. For example in 1) we have \( f(X) = X^4 + \overline{5} \in \mathbb{Z}_7[X] \), so \( \overline{5} \) here must mean \( [5]_7 \) since it lives in the coefficient ring \( \mathbb{Z}_7 \).

Section A

1 - Should be able to do this, but Dr Häsä has explicily said on duo that he would not ask this type of question.

Section B

7b - Notation: \( K^\times \) is different notation for \( K^\ast \), the units of \( K \).


2012

Notation: We would normally write \( \langle a_1, \ldots, a_n \rangle \), with angle brackets, for the ideal generated by \( a_1, \ldots, a_n \). Here it is written \( (a_1, \ldots, a_n) \), with round brackets.

Notation: \( \overline{a} \) is another way of writing cosets in a quotient ring (or group); the explicit notation can be worked out from context of the question. For example in 1iii) we have \( f(X) = X^5 + \cdots + \overline{5} \in \mathbb{Z}_3[X] \), so \( \overline{5} \) here must mean \( [5]_3 \) since it lives in the coefficient ring \( \mathbb{Z}_3 \).

Section A

1i, 1ii - Should be able to do this, but Dr Häsä has explicily said on duo that he would not ask this type of question.

2ii, iii, iv - Ideals with multiple generators are only mentioned briefly in the notes, at the end of section 12.1. Techniques for showing equality of ideals by manipulating generators has not really been discussed.

6i - Similar comment to 2014 question 6ii, you need to know the classification of groups of order \( 2p \) from Sheet 19 question 10. And the fact that all groups of order \( p^2 \) are abelian.

6ii - Writing down the groups should be fine, but counting orders of elements could be tricker. A sysematic method for doing this was discussed in lectures in previous years.

Section B

8 - Ideals with multiple generators are only briefly mentioned in the notes, at the end of Section 12.1

9e - The idea here is basically the same as in 2014 question 6ii, and the theorem on groups of order \( p^2 \).

10b - For us, \( D_n \) is defined as a permutation group. For this question notice that we have to find \( D_n \) as a subgroup of \( S_{2n} \), not just \( S_n \). This is most naturally done using Cayley's theorem since Cayley realises \( G \) as a subgroup of \( S_{\left\lvert G \right\rvert} \), and \( \left\lvert D_n \right\rvert = 2n \).


2011

Notation: We would normally write \( \langle a_1, \ldots, a_n \rangle \), with angle brackets, for the ideal generated by \( a_1, \ldots, a_n \). Here it is written \( (a_1, \ldots, a_n) \), with round brackets.

Nottion: \( \overline{a} \) is another way of writing cosets in a quotient ring (or group); the explicit notation can be worked out from context of the question.

Section A

1 - From the context of our course this question doesn't make sense, since we necessarily require a ring homomorphism to send \( 1_R \mapsto 1_S \). If you attempt this question just drop that requirement from a ring homomorphism, and only require \( \phi(a + b) = \phi(a) + \phi(b) \) and \( \phi(ab) = \phi(a)\phi(b) \).

6 - Modules were not discussed in any real detail, only briefly in the sketch of the proof of the classification of finitely generated abelian groups (Section 20.2 of Epiphany notes), and in questions 14 and 16 from Sheet 13. But you could give it a try.

Section B

7b, c - Prime ideals have not really been defined in the course, therefore part b is difficult to answer. At the end of section 13.1, it is stated that an ideal \( \mathfrak{p} \) in a commutative ring \( R \) is prime if the quotient \( R/\mathfrak{p} \) is an integral domain. This follows from the definition that an ideal \( \mathfrak{p} \) is prime if \( ab \in \mathfrak{p} \) implies \( a \in \mathfrak{p} \) or \( b \in \mathfrak{p} \).

Since part c) uses the answer to part b), and the classification of prime ideals in \( \mathbb{Z} \), this is also difficult to answer. One must use that the prime ideals in \( \mathbb{Z} \) are \( \langle 0 \rangle \), and \( \langle p \rangle \), for \( p \) a prime number in \( \mathbb{Z} \).

9b, c - These depend on using the class equation of a group to show that \( p \) divides the size of the centre \( Z(G) \). For part c, see 2014 question 6ii) and the theorem on groups of order \( p^2 \) following it.


2010

Notation: We would normally write \( \langle a_1, \ldots, a_n \rangle \), with angle brackets, for the ideal generated by \( a_1, \ldots, a_n \). Here it is written \( (a_1, \ldots, a_n) \), with round brackets.

Notation: \( \overline{a} \) is another way of writing cosets in a quotient ring (or group); the explicit notation can be worked out from context of the question.

Section A

1a - For us, all rings have identity, so we would not normally bother to say "with identity".

3 - Safe to ignore. Like 2014 question 3, this requires the full abstract Chinese remainder theorem, which has not been covered this year. Moreover, part b requires knowing the systematic way to go backwards. This has been covered in ENTC but maybe only for \( \mathbb{Z}_n \).

4b - This requires the classification of gorups of order \( 2p \) from Sheet 19 question 10.

4c - Systematic ways of counting orders of elements were developed in previous years, so this may be difficult to answer. You could still try it.


2009

Notation: We would normally write \( \langle a_1, \ldots, a_n \rangle \), with angle brackets, for the ideal generated by \( a_1, \ldots, a_n \). Here it is written \( (a_1, \ldots, a_n) \), with round brackets.

Secion A

1 - More sophisticated tests for polynomial irreducibility were developed in previous years. For i) you need Eisensein's Criterion, for ii) the Rational Root Theorem. Can use this in iii).

3 - Safe to ignore. Needs the chinese remainder theorem, and the way to go backwards as in 2010 question 3.

4 - Safe to ignore. Burnside's counting theorem was not covered. It is mentioned on Sheet 18 question 12f, but you are told to consult the internet. I'd recommend you read about it since it shows a very nice application of algebra (specifically group theory) to combinatorics -- you can now count things and disregard symmetricaly equivalent possibilities.

5b - Probably safe to ignore. Explicit methods to write an abelian group given in terms of generators and relations, as a product of \( \mathbb{Z} \)'s and \( \mathbb{Z}_n \)'s were given in previous years. These ideas are hinted at in the sketch of the proof of the classification of finitely generated abelian groups (Section 20.2 of Epiphany notes)

Section B

9iib) Safe to ignore. Sylow theorems have not been covered in the course. The first Sylow theorem is given in Sheet 18 challenging question 15, and but is not sufficient for this.

10ib) Sylow \( p \)-subgroup has not been defined in the course. The notion is developed in Sheet 18, question 15, but not named as such. For a finite group of order \( n \), it is a subgroup of order \( p^k \) where \( p^k \mid n \) but \( p^{k+1} \not\mid n \), which is shown to exist by the first Sylow theorem.