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Field theory

  1. Jan 23, 2008 #1
    [SOLVED] field theory

    1. The problem statement, all variables and given/known data
    Is the following sentence true:

    A field is a ring with nonzero unity such that the set of nonzero elements of F is a group under multiplication.

    2. Relevant equations

    3. The attempt at a solution
    I think it is false. To make it correct, we must require that the ring be commutative.
    Am I correct?
  2. jcsd
  3. Jan 23, 2008 #2
    If you're asking whether that definition implies commutativity, then you're correct, it does not. For example, you can take the quaternions with rational coefficients. They form a noncommutative field.

    If you're asking whether the definition of a field requires commutativity, it depends. Most people would define it that way, but some texts (for example, Serre's books) do not. Normally, one calls a noncommutative field a skew field or a division ring.
  4. Jan 23, 2008 #3
    Here is the definition of a field in my book (Farleigh):

    "Let R be a ring with unity 1 not equal to 0. An element u in R is a unit of R if it has a multiplicative inverse in R. If every nonzero element of R is a unit, then R is a division ring. A field is a commutative division ring."

    Please confirm the truth of the following statements:

    1)The nonzero elements of a field form a group under the multiplication in the field

    2)A commutative ring with nonzero unity such that the set of nonzero elements of F is a group under multiplication is also a field.
  5. Jan 24, 2008 #4
    1) yes
    2) yes
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