Fields and Algebraic Equations

Academic year
Subject Area
Language of Instruction
Mode of Delivery
ECTS Credits
1st Cycle Studies

Recommended Prerequisites

Linear Algebra and Analytic Geometry I, II; Groups and Symmetries

Teaching Methods

Classes are mainly expository but students are frequently invited to answer to small questions that have to be clarified in the context of proofs. Before the presentation of some extended technical topics, the instructor will give an overview of how the various results will be connected. Examples and exercises for applying the acquired knowledge will be included.

During the semester students can use tutorial time to clarify their difficulties in grasping the theory and in gaining practical knowledge

Learning Outcomes

We aim to convey some history and technical knowledge on the topic of  solving algebraic equation and on the history of algebra in general.  Another goal of the course unit is to show students the beauty and elegance of a mathematical theory through the understanding of abstract structures' role. At the end, students shoud have developed facility to solve algebric problems, acquired abstract thinking and should be able to write clearly statements and proofs

Work Placement(s)



 Short historical review of the search for solutions to algebraic equations and the reiterated extension of the number concept: the fundamental theorem of algebra;  the equations of the third and fourth degrees; the contributions of  Lagrange, Ruffini, Abel and Galois.

Field extensions. Applications: the field of constructible numbers and  unsolvability of the classical problems; the constructability of the 17-gon.

The fundamental theorem of Galois theory. The splitting field of a family of polynomials. The Galois group of a polynomial. Some results about separability, cyclic, cyclotomic and radical extensions. Applications: solving polynomial equations by radicals

Head Lecturer(s)

Jorge Manuel Senos da Fonseca Picado

Assessment Methods

Exam(90%) +Problem Resolving Report(10%) or MidtermExam (70%) + Test (30%)+Problem resolving report (10%): 100.0%


I. Stewart, Galois Theory,  Chapman and Hall, 2004.


I. Stewart, Why Beauty is Truth: a history of symmetry, Basic Books, 2007.


J. Fauvel (ed.) :  History of Mathematics, Histories of Problems, Ch.XII, IREM comission, 1997.


T. Hungerford, Algebra, GTM 73, Springer 1974.


How science can inform the classroom: Teachers need a trusted source to tell fads and fallacies from proved methods,

Scientific American,  September, 2012.