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Math Help - More ring homoorphism

  1. #1
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    More ring homoorphism

    Prove that phi:Z_p mapped to Z_p, phi(a)=a^p is a ring homomorphism and find ker phi.
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  2. #2
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    Quote Originally Posted by chadlyter View Post
    Prove that phi:Z_p mapped to Z_p, phi(a)=a^p is a ring homomorphism and find ker phi.
    phi(xy)=(xy)^p=(x^p)*(y^p) ---> It is a commutative ring.

    phi(x)phi(y)=(x^p)*(y^p).

    Thus, phi(xy)=phi(x)*phi(y)

    ---

    phi(x+y)=(x+y)^p=x^p+px^{p-1}y+...+pxy^{-1}+y^p
    Note that all term in middle of binomial expansion are divisible by p. Hence, this simplifies to x^p+y^p

    Thus,
    phi(x+y)=phi(x)+phi(y)

    Note,
    ker(phi) = {0}
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