Examinable material for Galois Theory - Neil Strickland

Examinable material for Galois Theory
General type of questions
As in previous years, a large proportion of the questions will ask you to apply the general machinery of
Galois theory to certain specific fields, especially cyclotomic fields, multiquadratic fields, cubic extensions
of Q, or unusually nice quartic extensions of Q. Another common type of question asks you to consider a
Galois extension where you are given certain information about groups and intermediate fields, and you need
to deduce other information using the Galois correspondence. You should review all the questions of these
types from the past papers and the notes.
Compared to exams that you have taken in previous years, there will be more places where you need some
creativity or ingenuity to find the right approach.
Background from earlier courses
The early sections of the notes are reminders of material that you should have covered in earlier courses.
Specifically:
• Almost all of Section 1 (Fields: definitions and examples) is covered by MAS333 (Fields).
• Almost all of Section 2 (Vector spaces) is covered by MAS277 (Vector spaces and Fourier Theory).
• The material in Section 3 (Ideals and quotient rings) is partly covered by MAS276 (Rings and
Groups), and also appears in MAS333 (Fields), MAS337 (Rings and Modules) and MAS434 (Commutative
Algebra) in varying levels of generality.
• Most of the material in Sections 4 and 5 should also be familiar from MAS333 (Fields).
The exam will not focus on any of this, but you will need it as background knowledge.
Omitted sections
In 2011-12, Sections 9 (Finite fields), 13 (Quartics), 14 (Cyclic extensions) and 15 (Extension by radicals)
were not covered and will not be examined.
I will not expect you to remember complex formulae from Section 12 (Cubics), but I will expect you to
understand the general method.
Theorems and other results
In the exam you will be asked to prove various results, which will be taken from the following list:
• Proposition 1.25 (automorphisms of Q)
• Proposition 1.29 (homomorphisms are injective)
• Proposition 1.31 (L H is a subfield)
• Proposition 4.42 (distinct roots for irreducible polynomials)
• Proposition 5.10 (algebraic extensions)
• Proposition 6.3 (Galois groups are groups)
• Proposition 7.2 (biquadratic extensions)
• Lemma 7.7 (subgroups of Σ p )
• Corollary 7.8 (Galois group is all of Σ p )
• Proposition 8.11 (Galois groups of cyclotomic fields)
• Proposition 12.1 (Cubic extensions).
There will also be unseen problems in which you need to find your own proofs of results that are not in
the notes. Moreover, you will need to understand and be able to use many other results. In particular, you
will need to know all the details of Theorem 11.1 (the Galois correspondence) and related results, but you
will not be asked to prove them.
1

Definitions
You will be asked to give various definitions, which will be taken from the following list:
• Definition 1.13 (subfield)
• Definition 1.19 (homomorphism, isomorphism, automorphism)
• Definition 2.18 (degree of an extension)
• Definition 4.11 (irreducible polynomial)
• Definition 4.15 (Eisenstein criterion)
• Definition 5.5 (algebraic extension)
• Definition 5.13 (split polynomial)
• Proposition 6.3 (Galois group)
• Definition 6.8 (normal extension)
• Definition 8.1 (cyclotomic polynomial).