I am giving tutorials for the course Teoria de Galois in the second semester 2009/2010 in the Departamento de Matematica da Faculdade de Ciencias da Universidade de Lisboa. 

Tutorial time: Thursday 9am-11am in Room 6.2.45.

Office Hours: Thursday 3pm-4pm.

A summary of the material that has been covered in the tutorials is given below.

Atenção

Since it is a public holiday there will NOT be a tutorial on Thursday 3rd June.

Instead there will be a tutorial on Monday 31st May, 11:00-13:00, in Room 6.2.45.

Tutorial 1, 23rd February 2010

We recalled the definition of reducible polynomial, and some basic tools for testing irreducibility including: the rational root test, Gauss's Lemma and Eisenstein's criterion.

We looked at the following exercises from Brison's book:

Chapter 1: 7 (a), (b), (c), (f), (g); 8 (a), (d), (h), (i), and I outlined the solution to the first part of 5 (full details may be found in the back of Brison's book).

Tutorial 2, 4th March 2010

We recalled the definition of field extension, degree of a field extension, simple extension, algebraic extension, the minimal polynomial, and results for working with field extensions including the Tower Theorem and Theorem 2.4 from Brison which shows that the degree of a simple extension is the degree of the minimal polynomial, and gives a basis for the extension.

We applied these results to solve the following exercises from Brison's book:

Chapter 2: 6, 7(a), 8 (solution discussed), 10, 11 (a), (b), (d) and (g).

Tutorial 3, 11th March 2010

We recalled the notions of K-isomorphism, monomorphism and automorphism, and Theorems 2.4 and 2.14 from Brison's book stating that simple algebraic extensions with the same minimal polynomial are isomorphic (and the converse).

We applied these results to solve the following exercises from Brison's book:

Chapter 2: 13 (a), 14 (a), (b), (c), (d) and (e).

Tutorial 4, 18th March 2010

We recalled the definitions of splitting field, normal extension and the normal closure of a field extension, and the results 1.10, 1.11, 1.13, and 1.15 from Chapter 3 of Brison's book.

We applied these results to solve the following exercises:

Chapter 3: 2, 5 (a), (b), (c), (d), (e), (f), and 1 (a), (b).

Tutorial 5, 25th March 2010

We recalled the definitions of splitting field, normal extension, normal closure, separable extension, and the results 1.10, 1.11, 1.13, 1.15, 2.4, and 2.7. from Chapter 3 of Brison's book.

Applying these results we worked through the following exercises:

Chapter 3: 1 (c), (d), (e), (f), (g), (h), (i), (j), and (k).

Tutorial 6, 8th April 2010

This tutorial was given by Dr Catarina Carvalho. Details may be found here.

Tutorial 7, 15th April 2010

We recalled the definitions of Galois extension and Galois group, and results III 1.10, III 1.13, III 2.7, V 1.4, V 1.5, and V 1.6 from Brison's book.

Applying these results we looked at the following exercises concerned with computing Galois groups:

Chapter 5: 1 (a) (c) (e) (f) and (j).

Tutorial 8, 22nd April 2010

We worked through questions from the "Teste-tipo de Teoria de Galois" from the module webpage which may be found by following the links from here.

Tutorial 9, 29th April 2010

We worked through question III from the Galois Theory class test, which shows how the Fundamental Theorem of Galois Theory can be applied.

Tutorial 10, 6th May 2010

The questions in this weeks tutorial were Q3 and Q5 from Chapter 5 of Brison's book, both of which are solved using the Fundamental Theorem of Galois Theory.

Tutorial 11, 20th May 2010

We recalled some basic facts about roots of unity, cyclotomic polynomials, and Euler's totient function. We the applied these results to answer questions 2,3,4,6(a),(b) and (c) from this question sheet.

Tutorial 12, 27th May 2010

We worked through the solution of 6(c) from this question sheet. We recalled the definition of the discriminant of a polynomial, and the formula for the discriminant of a cubic polynomial. We then proved a result classifying the possible Galois groups of a cubic polynomial f(t) over Q, in terms of the irreducibility of f(t), and whether or not the discriminant is a perfect square in Q; see Q16 from this question sheet. We applied this result to answer Q13 of this question sheet and discussed how the same result could be applied to answer questions 14(a), (b) and (d).

Tutorial 13, 31st May 2010

We recalled the definitions of soluble group, radical field extension, and what it means for a polynomial to be soluble by radicals. We recalled Cauchy's theorem (Brison IV.7.6), Galois's Theorem (Brison VI.2.10), and the fact that the symmetric group on p points (where p is prime) is generated by any transposition and p-cycle. Using these definitions and results we showed that the polynomial:

f(t) = t^5 - 4t^2 + 2

is not soluble by radicals over Q. Using the same approach we solved the exam question [2008(3)]Q5. We discussed the fact that the same methods can be used to solve all of the following exam questions: [2007(1)]Q5, [2006(2)]Q4, [2006(1)]Q5, [2003(2)]Q6.

Galois Theory Past Papers

Here are copies of some previous exam papers:

[2001(1)] [2001(2)] [2001(3)] [2001(4)] [2002(1)] [2002(2)] [2002(3)] [2002(4)] [2003(1)] [2003(2)] [2003(3)] [2003(4)] [2004(1)] [2006(1)] [2006(2)] [2006(3)] [2007(1)] [2007(2)] [2007(4)] [2008(1)] [2008(2)] [2008(3)]