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Math Help - Surjective Ring Homomorphism

  1. #1
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    Surjective Ring Homomorphism

    Hi there!

    This is my first time posting here so apologies if this is in the wrong section or there's already a post about this but I have looked everywhere to try and get a solution to this.

    The question I'm stuck on is:

    Let θ : Q[X] → Q(√3) be the map defined by θ(a0 + a1X + ... + anXn) = (a0 + a1√3 + ... + an(√3)n).
    Show that θ is a surjective ring homomorphism.
    Prove that Ker θ = (X^2 − 3)Q[X].
    Deduce that the factor ring Q[X]/Ker θ is isomorphic to Q(√3).

    The notes I have on this aren't particularly useful though I do believe that I need to use the first isomorphism theorem for the last part. The first part however I can't seem to find anything like in any of the books or websites I have looked at.

    Thanks in advance for any help you can provide.
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  2. #2
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    One example I have found seems to be relevant but unhelpful as it doesn't really explain each step properly.

    φ: Q[x] -> Q(√2)
    f(x) = ∑aiX^i - > φ(f(x)) = ∑ai(√2)^i = f(√2)

    φ is a surjective homomorphism: if a + b√2 ∈ Q(√2), let f(x) = a + bX, then φf(x) = f(√2) = a + b√2.

    Claim: Ker φ = (X^2 - 2)Q[x] = I

    Which then goes on to find Ker φ.

    Does anyone know the steps for Q(√3)?
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  3. #3
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    Nov 2008
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    Quote Originally Posted by Brattmat View Post
    Hi there!

    This is my first time posting here so apologies if this is in the wrong section or there's already a post about this but I have looked everywhere to try and get a solution to this.

    The question I'm stuck on is:

    Let θ : Q[X] → Q(√3) be the map defined by θ(a0 + a1X + ... + anXn) = (a0 + a1√3 + ... + an(√3)n).
    Show that θ is a surjective ring homomorphism.
    Prove that Ker θ = (X^2 − 3)Q[X].
    Deduce that the factor ring Q[X]/Ker θ is isomorphic to Q(√3).

    The notes I have on this aren't particularly useful though I do believe that I need to use the first isomorphism theorem for the last part. The first part however I can't seem to find anything like in any of the books or websites I have looked at.

    Thanks in advance for any help you can provide.
    Verify that \theta is a ring homomorphism defined by q \mapsto q for each q \in \mathbb{Q} and x \mapsto \sqrt{3}, otherwise. An ideal ( x^2-3) is clearly kernel of \theta since \theta(I)=0 for I \in (x^2-3) . Now, you apply the first isomorphism theorem and obtain the result.
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