Results 1 to 2 of 2

Math Help - modulo squares and cubes

  1. #1
    Newbie
    Joined
    Dec 2009
    Posts
    1

    modulo squares and cubes

    If we look at the field F_7* modulo the squares, then we can say that this is isomorphic to Z/2 by sending the squares to 0, and the non-squares to 1 (at least we think this is the reason that it is isomorphic to Z/2).

    Now we want to look at F_7* modulo cubes, we think this is isomorphic to Z/3, but we don't know why. We can send the cubes to 0, but how do we know which elements are mapped to 1 and 2?

    We can't find anything about this on the web...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by lizzy View Post
    If we look at the field F_7* modulo the squares, then we can say that this is isomorphic to Z/2 by sending the squares to 0, and the non-squares to 1 (at least we think this is the reason that it is isomorphic to Z/2).

    Now we want to look at F_7* modulo cubes, we think this is isomorphic to Z/3, but we don't know why. We can send the cubes to 0, but how do we know which elements are mapped to 1 and 2?

    We can't find anything about this on the web...



    Since a^3\in\{-1,0,1\}\!\!\pmod 7 , then a^3\in\{-1,1\}\!\!\pmod 7 for a\in\left(\mathbb{F}_7\right)^* . Also, if \phi: \left(\mathbb{F}_7\right)^*\rightarrow \mathbb{Z}_3 is such a homomorphism, it must be that \phi(ab)=\phi(a)+\phi(b) ,

    where the product in the left side is modulo 7, whereas the sum in the right side is modulo 3.

    But it must be \phi(a)=\phi(b)\!\!\!\pmod 3\Longleftrightarrow \phi(a)-\phi(b)=0\!\!\!\pmod 3  \Longleftrightarrow \phi(ab^{-1})=0\!\!\!\pmod 3\Longleftrightarrow ab^{-1}=\pm 1\!\!\!\pmod 7 \Longleftrightarrow a=\pm b\!\!\!\pmod 7 .

    From here you can now build your function \phi taking into account, of course, that it must be \phi(\pm 1)=0 since only \pm 1 are cubes in \left(\mathbb{F}_7\right)^*...

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Modulo of squares = modulo of roots
    Posted in the Discrete Math Forum
    Replies: 4
    Last Post: December 1st 2009, 09:04 AM
  2. Replies: 4
    Last Post: November 13th 2009, 05:12 PM
  3. Of Squares & Cubes
    Posted in the Number Theory Forum
    Replies: 5
    Last Post: May 5th 2009, 07:53 PM
  4. cubes and squares
    Posted in the Algebra Forum
    Replies: 2
    Last Post: February 2nd 2009, 08:53 AM
  5. two cubes
    Posted in the Algebra Forum
    Replies: 1
    Last Post: February 19th 2007, 07:03 AM

Search Tags


/mathhelpforum @mathhelpforum