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Field Extensions: Field Extension

Field Extensions: Field Extension


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About the Book

Purchase includes free access to book updates online and a free trial membership in the publisher's book club where you can select from more than a million books without charge. Excerpt: In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties. Field extensions can be generalized to ring extensions which consist of a ring and one of its subrings. Let L be a field. If K is a subset of the underlying set of L which is closed with respect to the field operations and inverses in L, then by definition K is a subfield of L, and L is an extension field of K. L/K, read as "L over K," is a field extension. If L is an extension of F which is in turn an extension of K, then F is an intermediate field (or intermediate extension or subextension) of the field extension L/K. Given a field extension L/K and a subset S of L, K(S) denotes the smallest subfield of L which contains K and S, a field generated by the adjunction of elements of S to K. If S consists of only one element s, K(s) is a shorthand for K({s}). A field extension of the form L = K(s) is called a simple extension and s is called a primitive element of the extension. Given a field extension L/K, then L can also be considered as a vector space over K. The elements of L are the "vectors" and the elements of K are the "scalars," with vector addition and scalar multiplication obtained from the corresponding field operations. The dimension of this vector space is called the degree of the extension, and is denoted by . An extension of degree 1 (that is, one where L is equal to K) is called a trivial extension. Extensions of degree 2 and 3 are called quadratic extensions and cubic extensions, respectively. Depending on whether the degree is finite or infinite the extens... More: http: //booksllc.net/?id=11634


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Product Details
  • ISBN-13: 9781156216064
  • Publisher: Books LLC
  • Publisher Imprint: Books LLC
  • Height: 152 mm
  • Sub Title: Field Extension
  • Width: 229 mm
  • ISBN-10: 1156216060
  • Publisher Date: 31 May 2010
  • Binding: Paperback
  • Spine Width: 3 mm
  • Weight: 73 gr


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