Algebraic Closure - Examples

Examples

  • The fundamental theorem of algebra states that the algebraic closure of the field of real numbers is the field of complex numbers.
  • The algebraic closure of the field of rational numbers is the field of algebraic numbers.
  • There are many countable algebraically closed fields within the complex numbers, and strictly containing the field of algebraic numbers; these are the algebraic closures of transcendental extensions of the rational numbers, e.g. the algebraic closure of Q(π).
  • For a finite field of prime order p, the algebraic closure is a countably infinite field which contains a copy of the field of order pn for each positive integer n (and is in fact the union of these copies).
  • See also Puiseux expansion.

Read more about this topic:  Algebraic Closure

Famous quotes containing the word examples:

    “In the examples that I here bring in of what I have [read], heard, done or said, I have refrained from daring to alter even the smallest and most indifferent circumstances. My conscience falsifies not an iota; for my knowledge I cannot answer.”
    —Michel de Montaigne (1533–1592)

    “No rules exist, and examples are simply life-savers answering the appeals of rules making vain attempts to exist.”
    —André Breton (1896–1966)

    “Histories are more full of examples of the fidelity of dogs than of friends.”
    —Alexander Pope (1688–1744)