Results 1 to 2 of 2

Math Help - Solutions in commutative ring.

  1. #1
    Junior Member
    Joined
    Jan 2010
    Posts
    39

    Solutions in commutative ring.

    Show that the equation x^{129}+x+2=0 has precisely three solutions in the commutative ring Z_{256}.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Banned
    Joined
    Oct 2009
    Posts
    4,261
    Thanks
    2
    Quote Originally Posted by featherbox View Post
    Show that the equation x^{129}+x+2=0 has precisely three solutions in the commutative ring Z_{256}.

    The multiplicative group of units of that ring is U:=U(\mathbb{Z}_{256}) whose order is \phi(256=2^8)=2^7=128\Longrightarrow u^{128}=1\,\,\,\forall\,u\in U

    \Longrightarrow \,\forall x\in U\,,\,\,x^{129}+x+2=2x+2=2(x+1)=0\!\!\!\pmod {256} \iff x=-1\!\!\!\pmod{256} or x+1=128\!\!\!\pmod {256}\iff x=127,\,255\!\!\!\pmod{256} , and

    these are the only solutions with x\in U


    No, if x=2k\notin U is a solution then x^{129}+x+2=2^{129}k^{129}+2k+2=\left(2^8\right)^{  16}\cdot 2k^{129}+2k+2 =2(k+1)=0\!\!\!\pmod {256}\iff k+1=128\iff k=127 and

    then x=254.

    Tonio
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Commutative Ring
    Posted in the Advanced Algebra Forum
    Replies: 5
    Last Post: March 10th 2011, 08:47 AM
  2. Commutative Ring
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: February 22nd 2011, 11:38 AM
  3. Commutative ring
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: April 8th 2010, 07:05 AM
  4. Commutative Ring
    Posted in the Advanced Algebra Forum
    Replies: 4
    Last Post: November 24th 2009, 09:54 AM
  5. Non-commutative ring
    Posted in the Advanced Algebra Forum
    Replies: 9
    Last Post: March 22nd 2009, 03:12 AM

Search Tags


/mathhelpforum @mathhelpforum