Results 1 to 4 of 4

Math Help - ring homomorphism

  1. #1
    Newbie
    Joined
    Oct 2008
    From
    Punjab
    Posts
    23

    Groups prob. Help needed

    1)consider Z24 as additive group modulo 24.Then find the number of element of order 8 in group 24 ?

    2)Consider Z5 and Z20 as ring modulo 5 and 20.Them the number of homomorphisms
    p:Z5-->Z20 is?

    3)if Q is a field of rational numbers and Z2 as a field of modulo 2 let
    f(x)=x^3-9x^2+9x+3 then witch of following is correct.1)f(x) is irreducible over Z2
    2)f(x) is irreducible over both Q and Z2
    3)f(x) is reducible over Q but irreducible and Z2
    4)f(x) is reducible over both Z2 and Q

    please explain with detail.Thanks in advance
    (these questions asked by indian institue of technology delhi asked in gate test )
    Last edited by reflection_009; October 9th 2008 at 08:54 AM. Reason: getting very less answers
    Follow Math Help Forum on Facebook and Google+

  2. #2
    is up to his old tricks again! Jhevon's Avatar
    Joined
    Feb 2007
    From
    New York, USA
    Posts
    11,663
    Thanks
    3
    Quote Originally Posted by reflection_009 View Post
    1)consider Z24 as additive group modulo 24.Then find the number of element of order 8 in group 24 ?
    this is asking for the number of solutions to [8a]_{24} = [0]_{24}. that is, how many solutions are there to the congruence equation 8a \equiv 0~(\text{mod }24)
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Junior Member
    Joined
    Oct 2008
    Posts
    64
    Quote Originally Posted by reflection_009 View Post
    1)

    3)if Q is a field of rational numbers and Z2 as a field of modulo 2 let
    f(x)=x^3-9x^2+9x+3 then witch of following is correct.1)f(x) is irreducible over Z2
    2)f(x) is irreducible over both Q and Z2
    3)f(x) is reducible over Q but irreducible and Z2
    4)f(x) is reducible over both Z2 and Q
    Since f is of degree one, f is reducible if and only if f has a linear factor.
    Now recall that x-c is a linear factor if f(c)=0. So basically we only need to check if there is an element c in the field that make f(c)=0
    for Z2, we have f(0),f(1) not equal zeero so f is irreducible there. For Q, by Gauss Lemma we need only to check if f has an integer root. This integer root must divides the constant term, i.e. 3. So you need to check f(1),f(-1),f(3),f(-3). If one of them is zero then it is reducible but if not then it is irreducible.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    10
    Quote Originally Posted by reflection_009 View Post
    1)consider Z24 as additive group modulo 24.Then find the number of element of order 8 in group Z24 ?
    Let 3\mathbb{Z}_{24} = \left< [3] \right>. This is a subgroup of order 24/3 = 8. Any element of order 8 must lie in this subgroup. Since 3 is a generator it means all other generators i.e. elements of order 8 are k[3] where \gcd(k,8)=1. Therefore the are \phi(8) such elements.

    2)Consider Z5 and Z20 as ring modulo 5 and 20.Them the number of homomorphisms
    p:Z5-->Z20 is?
    Hint: If \phi is a homomorphism then it is completely determined by its value on \phi ([1]_5]). Find the possibilities that this can be.

    3)if Q is a field of rational numbers and Z2 as a field of modulo 2 let
    f(x)=x^3-9x^2+9x+3 then witch of following is correct.1)f(x) is irreducible over Z2
    2)f(x) is irreducible over both Q and Z2
    3)f(x) is reducible over Q but irreducible and Z2
    4)f(x) is reducible over both Z2 and Q
    x^3 - 9x^2+9x+3 is irreducible over \mathbb{Z}_2 if and only if it has no zeros in \mathbb{Z}_2. Likewise for \mathbb{Q}.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Ring Homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: March 10th 2011, 08:49 AM
  2. Ring homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: August 7th 2010, 09:22 AM
  3. homomorphism ring ~
    Posted in the Advanced Algebra Forum
    Replies: 1
    Last Post: December 23rd 2009, 11:34 AM
  4. Ring homomorphism
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: May 12th 2009, 05:50 AM
  5. Is this a ring homomorphism?
    Posted in the Advanced Algebra Forum
    Replies: 2
    Last Post: April 18th 2009, 10:28 AM

Search Tags


/mathhelpforum @mathhelpforum