Galois Theory
By Emil Artin
* Publisher: University of Notre Dame Press
* Number Of Pages: 82
* Publication Date: 1959
* ISBN-10 / ASIN: 0268001081
* ISBN-13 / EAN: 9780268001087
Product Description:
Clearly presented elements of one of the most penetrating concepts in modern mathematics include discussions of fields, vector spaces, homogeneous linear equations, extension fields, polynomials, algebraic elements, as well as sections on solvable groups, permutation groups, solution of equations by radicals, and other concepts. 1966 ed.
Galois Theory is in traditional mathematical format. The major elements of the book are definitions, lemmas, theorems, and proofs. The book introduces the major topics of Galois Theory. They are fields, extension fields, splitting fields, unique decomposition of polynomials into irreducible factors, solvable groups, permutation groups, and solution of equations by radical.
The last part of the book contains the major results of Galois Theory with proofs using the theorems from the second part of the book. They are theorem 5: The polynomial f(x) is solvable by radicals if and only if its group is solvable; theorem 4: The symmetric group G on n letters is not solvable for n > 4; theorem 6: The group of the general equation of degree n is the symmetric group on n letters. The general equation of degree n is not solvable by radicals if n > 4.
This is my second Galois Theory book. What impress me most is the involvement to prove the major results of Galois Theory such as theorem 5 and theorem 6. In order to prove the theorems, mathematicians invent many mathematical objects. They are root, group, symmetric group, solvable group, field, extension field, splitting field, Kummer field/extension, Abelian group, normal subgroup, normal extension, factor/quotient group, homomorph, fixed field, extension by radicals field, and more. Nowadays, we put all these objects under the domain of abstract algebra.
The book is certainly not self-contained because one would need an abstract algebra textbook for reference to the mathematical objects.
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