Tutorials, Hilary term, week 2

Here are some remarks on problem 7 (i) of Linear Algebra II, sheet 1. Let me know if you have a simpler proof.

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Here are some further remarks. Recall that the key statement is this:

For if

then

It is perhaps better to write it in the form

This is giving a lower bound for the number of cycles in , saying that if is small, then has to be big. I hope this makes intuitive sense: with just a few transpositions, you can’t create long cycles. Hence, there has to be many of them.

Note also that the strategy of proof (I learned from Matei’s answer) changes the perspective of the problem in a nice way. Instead of trying to prove a lower bound for the number of transpositions given a cycle decomposition, one tries to prove a lower bound for the number of cycles, given an expression in terms of transpositions. Possibly, this is easier because the cycle decomposition is essentially unique.

Collections, Hilary Term, 2012

I’ve marked the scripts for the algebra portion of the Merton collections for Mods Pure Maths, Part A AC1 and Part A AC2.
I am including here brief remarks on the questions. Later, if I have the energy, I will also upload comments on the Banach Spaces paper for Part B students.

POSTECH notes, algebra.

Here are some notes on

Automorphisms of algebraic number fields

to accompany the lecture on 5 December, 2011.

POSTECH notes, topology of Q_p.

Topology of and

POSTECH notes on constructible numbers

Constructible numbers

Maximal ideals in Z[x]

Maximal ideals

Postech NT, homework 2

NT homework 2

Non-examinable material

I received the following message:

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Dear Professor,

In the list of non-examinable material in the book you put sections 1.1.3- 1.1.5. However section 1.1.6. seems to rely on those sections ruled out. I wasn’t sure if it was the case that you forgot to list it, or that this section is indeed examinable?
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I’m at home right now without the textbook, but since the examination is near, I thought I’d send a quick generic clarification. If material in section B depends on section A and the latter is listed as ‘non-examinable,’ this means that the results needed from section A should simply be assumed when reading section B. I’m sorry if that wasn’t clear.

I also apologize that I didn’t manage to schedule another revision session.

Roots and coefficients of polynomials

A question came up in the algebraic geometry revision session about the relation between the roots and coefficients of a polynomial. Here is a note that summarizes the statement.

Remark on the textbook

During the algebraic geometry revision session today, a question arose about the portions of the textbook that were not discussed carefully on class. So here is a list of the parts that are not examinable:

section 1.1.3
section 1.1.4
section 1.1.5
section 1.3.2

chapter 5
chapter 6

If there is a result in some of the other parts that depend on the results in the these sections, the *statement* used should still be regarded as important, even if the proof will not be examinable.

The introduction should be read, even if it is not examinable in a strict sense (except the ideas that reappear elsewhere).