1 / 10

Field Extension

Field Extension. The main study of Field Theory By: Valerie Toothman. What is a Field Extension?. Abstract Algebra Main object of study in field theory The General idea is to start with a field and construct a larger field that contains that original field and satisfies additional properties.

noel
Download Presentation

Field Extension

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Field Extension The main study of Field Theory By: Valerie Toothman

  2. What is a Field Extension? • Abstract Algebra • Main object of study in field theory • The General idea is to start with a field and construct a larger field that contains that original field and satisfies additional properties

  3. Definitions • Field - any set of elements that satisfies the field axioms

  4. Definitions • Subfield – Let L be a field and K be a subset of L. If the subset K of L is closed under the field operations inherited from L, then the subset K of L is a subfield of L. • Extension Field- If K is a subfield of L then the larger field L is said to be the extension field of K. • Notation – L/K (L over K) signifies that L is an extension field of K • Degree – The field L can be considered as a vector space over the field K. The dimension of this vector space is the degree denoted by [L:K]

  5. Example • The field of complex numbers C is an extension field of the field of real numbers R, and R in turn is an extension field of the field of rational numbers Q. • C- a+bi where a is real a number • R – includes all rational numbers • So we say C/R , R/Q, and C/Q

  6. Example • The set Q(√2) = {a + b√2 | a, b ∈ Q} is an extension field of Q. • Degree - √2 is a root of -2 which cannot be factored in Q[x] so we use {1, √2} as a basis. Therefore the degree is 2

  7. One Happy Family! Field Extension Algebraic Extension Finite Extension Galois Extension (Normal and Separable extension)

  8. Galois Theory • Galois theory- the study of algebraic extensions of a field. Algebraic extensions is a kind of field extension (L/K) that for every element of L is a root of some non-zero polynomial with coefficients in K. • In General it provides a connection between field theory and group theory by Roots of a given polynomial.

  9. The Theory of Field extensions (including Galois theory) • Leads to impossibility proofs of classical problems such as angle trisection and squaring the circle with a compass and straightedge

  10. Field Extensions The main study of Field Theory By: Valerie Toothman

More Related