Notes on Galois Theory
Sudhir R. Ghorpade
Department of Mathematics, Indian Institute of Technology, Bombay 400 076
Email : [email protected]
October 1994
Contents
1
Preamble
2
2
Field Extensions
3
3
Splitting Fields and Normal Extensions
6
4
Separable Extensions
9
5
Galois Theory
11
6
Norms and Traces
16
1
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1
Preamble
These notes attempt to give an introduction to some basic aspects of Field Theory and Galois
Theory.
Originally, the succeeding sections of these notes constituted a part of the notes
prepared to supplement the lectures of the author on Galois Theory and Ramification Theory
at the All India Summer School in Number Theory held at Pune in June 1991. Subsequently,
the first 6 sections of the Pune Notes were separated and slightly revised to form these “Notes
on Galois Theory”, which were used for preconference distribution to the participants of the
NBHM sponsored Instructional School on Algebraic Number Theory (University of Bombay,
December 1994) at the request of the organisers. A few minor revisions have taken place in
the subsequent years.
The main aim of these notes has always been to provide a geodesic, yet complete, presen
tation starting from the definition of field extensions and concluding with the Fundamental
Theorem of Galois Theory. Some additional material on separable extensions and a section on
Norms and Traces is also included, and some historical comments appear as footnotes. The
prerequisite for these notes is basic knowledge of Abstract Algebra and Linear Algebra not
beyond the contents of usual undergraduate courses in these subjects. No formal background
in Galois Theory is assumed. While a complete proof of the Fundamental Theorem of Galois
Theory is given here, we do not discuss further results such as Galois’ theorem on solvability
of equations by radicals. An annotated list of references for Galois Theory appears at the end
of Section 5. By way of references for the last section, viz., Norms and Traces, we recommend
Van der Waerden’s “Algebra” (F. Ungar Pub. Co., 1949) and Zariski–Samuel’s “Commutative
Algebra, Vol. 1” (SpringerVerlag, 1975).
It appears that over the years, these notes are often used by students primarily interested
in Number Theory. Thus it may be pertinent to remark at the outset that the topics discussed
in these notes are very useful in the study of Algebraic Number Theory
1
. In order to derive
maximum benefit from these notes, the students are advised to attempt all the Exercises
and fill the missing steps, if any, in the proofs given. The author would appreciate receiving
comments, suggestions and criticism regarding these notes.
1
In fact, questions concerning integers alone, can sometimes be answered only with the help of field ex
tensions and certain algebraic objects associated to them. For instance, Kummer showed that the equation
X
p
+
Y
p
=
Z
p
has no integer solution for a class of odd primes
p
, called regular primes, which include all odd
primes less than 100 except 37, 59 and 67. Even a convenient definition of regular primes, not to mention the
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 Math, Group Theory, Galois theory, finite field, Field extension

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