Field and Galois Theory: 01 Introduction, Field Extensions

Advanced Math by Professor Roman・2 minutes read

Professor Roman's abstract algebra course on Field Theory covers topics such as field extensions, lattice of subfields, and Galwa theory, delving into embeddings of fields and the concept of algebraic independence. The course also explores the fundamental theorem of algebra, roots of unity, and the concept of field extensions, emphasizing the degree of a field extension and the prime subfield's characteristics.

Insights

  • Prior knowledge of vector spaces, groups, and rings is essential before delving into Field Theory, making it challenging to start abstract algebra studies directly with this course.
  • The degree of a field extension is a fundamental concept in Field Theory, denoted as square brackets e colon F, and is multiplicative, showcasing the relationship between different fields within the theory.

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Recent questions

  • What does Professor Roman's fourth course in abstract algebra focus on?

    Field Theory

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Summary

00:00

"Field Theory: Advanced Abstract Algebra Course"

  • Professor Roman's fourth course in abstract algebra focuses on Field Theory, following previous courses on linear algebra, group theory, and ring theory.
  • Each course is accompanied by a textbook available on Professor Roman's website, www.sroman.com, to aid in following along with the lectures.
  • Field Theory requires prior knowledge of vector spaces, groups, and rings, making it challenging to start abstract algebra studies directly with this course.
  • The course covers topics such as field extensions, lattice of subfields, algebraic elements, separability, and various types of field extensions.
  • It delves into embeddings of fields, homomorphisms, and the concept of algebraic independence in fields.
  • Galwa theory is introduced, discussing symmetric polynomials, Newton's theorem, and the fundamental theorem of Galwa theory.
  • Roots of unity, binomials, and solvability by radicals are explored, including the famous theorem that roots of some polynomials cannot be computed by algebraic formulas.
  • The course concludes with a proof of the fundamental theorem of algebra and an appendix on Galwa's historical contributions.
  • Definitions and facts from the Ring Theory course are revisited, including integral domains, fields, characteristic of a ring, and classification of fields.
  • The concept of field extensions is explained, with terminology like base field, intermediate field, and tower of fields introduced for better understanding.

26:12

Polynomial Common Factors and Field Extensions

  • The greatest common divisor of two polynomials lies in the ring K square bracket X, with coefficients in the smallest subfield of F.
  • If two polynomials over F have a non-constant common factor in an extension E, they also have it in the smaller field F.
  • The existence of a non-constant common factor between polynomials is field-independent.
  • Polynomials are relatively prime if they have no non-constant common factors over any field containing their coefficients.
  • Every non-constant polynomial over a field F has a root in some extension of F.
  • The degree of a field extension is denoted as square brackets e colon F and is called the degree of e over f.
  • The degree of a field extension is multiplicative, with the degree of e over f being the product of the degrees of e over K and K over f.
  • The prime subfield of a field F is isomorphic to the rational numbers if the characteristic of F is zero, and to CP if the characteristic is prime P.
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