Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Symmetric bilinear forms

  1. Aug 4, 2013 #1
    When I start to read the the article called "symmetric bi-linear forms", I face the following sentence. But I don't understand what does the following sentence suggest. Could someone please help me here?

    We will now assume that the characteristic of our field is not 2 (so 1 + 1 is not = to 0)
     
  2. jcsd
  3. Aug 4, 2013 #2

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    A field is a very general object. It is a set ##F## together with two functions ##+:F\times F\rightarrow F## and ##\cdot:F\times F\rightarrow F## satisfying:

    • ##a+(b+c) = (a+b)+c##
    • ##a+b=b+a##
    • There exists a ##0\in F## such that ##a+0 = 0+a = a##
    • For each ##a\in F##, there is an element ##b\in F## such that ##a+b=b+a=0##
    • ##a\cdot(b\cdot c) = (a\cdot b)\cdot c##
    • ##a\cdot b = b\cdot a##
    • There exists an ##1\in F## such that ##1\neq 0## and ##a\cdot 1 = 1\cdot a = a##
    • If ##a\neq 0##, then there exists a ##b\in F## such that ##a\cdot b =b\cdot a = 1##

    The usual suspects, such as ##\mathbb{Q}##, ##\mathbb{R}## and ##\mathbb{C}## are fields. All of these fields satisfy that ##1+1+1+1+1+....+1## (n times) is nonzero. But this is not a general property of a field. Fields that have the property are said to be of characteristic 0.

    For example, consider ##F=\{0,1\}## and define

    [tex]1+0 = 0+1 = 1~\text{and}~1+1=0+0=0[/tex]

    and

    [tex]1\cdot 0 = 0\cdot 0 = 0\cdot 1 = 0~\text{and}~1\cdot 1 = 1[/tex]

    This satisfies all the field axioms, but it has ##1+1=0##. We say that this field has characteristic 2.

    In general, if a field satisfies ##1+1+1+...+1=0## (n times). Then the field is said to have characteristic ##n##. We can always show that ##n## is a prime number.

    The main advantage of assuming that the field is not of characteristic ##0##, is that we can divide by ##1+1##. If we use the nice notation ##2=1+1##, then ##1/2## exists.
     
  4. Aug 4, 2013 #3
    Could you please explain it a bit more?


    Specially the following.


    What do you mean by "+" and "."?
     
  5. Aug 4, 2013 #4

    micromass

    User Avatar
    Staff Emeritus
    Science Advisor
    Education Advisor
    2016 Award

    The cartesian product ##F\times F## just means the set of all ordered pairs. Thus

    [tex]F\times F = \{(a,b)~\vert~a,b\in F\}[/tex]

    For example ##(0,0)\in \mathbb{R}\times\mathbb{R}## and ##(-1,2)\in \mathbb{R}\times \mathbb{R}##.

    That ##f## is a function from ##F\times F## to ##F## just means that we associate with each element in ##F\times F##, an element in ##F##.

    So given ##(a,b)## with ##a,b\in F##, we associate an element ##f(a,b)\in F##. Specifically, if ##f## is ##+##, then to each ordered pair ##(a,b)## with ##a,b\in F##, we associate ##+(a,b)## in ##F##. We write this as ##a+b##.

    So to each two elements in ##F##, we just associate a new element in ##F##.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook




Similar Discussions: Symmetric bilinear forms
  1. Bilinear Form (Replies: 1)

  2. Bilinear forms (Replies: 1)

Loading...