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Math Help - fields and subfields

  1. #1
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    fields and subfields

    Assume Q[sqrt(2)] is a subfield of the Real field R, and prove that it is the smallest subfield of R that contains sqrt(2)

    Q[sqrt(2)]= a + b(sqrt(2))
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  2. #2
    is up to his old tricks again! Jhevon's Avatar
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    Quote Originally Posted by JCIR View Post
    Assume Q[sqrt(2)] is a subfield of the Real field R, and prove that it is the smallest subfield of R that contains sqrt(2)
    how is Q[\sqrt{2}] defined?
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    Quote Originally Posted by JCIR View Post
    Assume Q[sqrt(2)] is a subfield of the Real field R, and prove that it is the smallest subfield of R that contains sqrt(2)

    Q[sqrt(2)]= a + b(sqrt(2))
    The way \mathbb{Q}(\sqrt{2}) is usually defined is to be the smallest subfield containing \mathbb{Q} and \sqrt{2}. Look Here.

    So I will assume you are defining \mathbb{Q}(\sqrt{2}) = \{ a+b\sqrt{2}|a,b\in \mathbb{Q}\} and you want to show it is the smallest such field. Let \mathbb{Q}\subseteq K\subseteq \mathbb{R} be a field containing \mathbb{Q} and \sqrt{2}. We need to show \mathbb{Q}(\sqrt{2})\subseteq K. Let x\in \mathbb{Q}(\sqrt{2}) then x=a+b\sqrt{2} but then a+b\sqrt{2}\in K because it is closed under sums and products, thus, x\in K.
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    This is JCIR you answered my question on fields and subfields, can you tell my why is it closed under sums and products ( i need to prove that too)
    Thanks alot for your help.
    Because a field has to be closed under sums and products (that is basically the definition of "field"). Meaning if a,b\in F then a+b,ab\in F.
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