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Are these following exercises equivalent?

  1. Let $\mathbb R$ be an extension of $\mathbb Q$.Find a polynomial p(x) in $\mathbb Q$[x]-{$0$} such that p($\sqrt2$+$\sqrt3$)=$0$.

and

2.Find the minimal polynomial of $\sqrt2$+$\sqrt3$ over$\mathbb Q$[x]

Aaron Martinez
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  • I know the answer already exist in this site, but that's not what I'm asking @dxiv – Aaron Martinez Apr 26 '17 at 04:55
  • What is the distinguishing difference in your question then? – dxiv Apr 26 '17 at 04:56
  • that's just the title of the question @dxiv – Aaron Martinez Apr 26 '17 at 04:57
  • That part is answered in the body of the linked question ($u^4 -10u^2 +1=0$) along with the explanation of how it was derived. – dxiv Apr 26 '17 at 04:58
  • I am actually mildly surprised, if the linked thread is the first time this question has been handled. – Jyrki Lahtonen Apr 26 '17 at 05:00
  • so just to clarify, 1. is equivalent to 2.? @dxiv – Aaron Martinez Apr 26 '17 at 05:01
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    @AaronMartinez no. Take $m(x)$ to be the minimal polynomial (which is unique). Then for any polynomial $f(x)$, $m(x)f(x)$ satisfies 1. – Oiler Apr 26 '17 at 05:03
  • @AaronMartinez #1 asks for a polynomial, while #2 asks for the minimal polynomial. The two are not equivalent. Again, see the linked question and the answers under it. – dxiv Apr 26 '17 at 05:03
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    @dxiv got it, thanks. – Aaron Martinez Apr 26 '17 at 05:06
  • Strictly speaking, the present question does not ask for a minimal polynomial, only if two exercises are equivalent, meaning presumably whether an answer to either one of them yields immediately an answer to the other one. And the answer to that is: "No they are not (equivalent)" since an answer to 1. might not yield an answer to 2. – Did Apr 26 '17 at 07:24
  • @Did but the answer of the second question answers too the first question? – Aaron Martinez Apr 26 '17 at 14:24
  • @AaronMartinez Hmmm... if you have the slightest idea of the meaning of the words you use, you should not be asking this: yes, finding the minimal polynomial of $\alpha$ suffices to find a nonzero polynomial whose evaluation at $\alpha$ is zero. – Did Apr 26 '17 at 18:00
  • @Did I don't know how to feel about your answer haha :) – Aaron Martinez Apr 26 '17 at 18:38
  • Then focus on the mathematical definitions involved, this will be more productive than wondering how you feel about this and that. – Did Apr 26 '17 at 22:19
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    :( you're right man, I'll take it as an advice.I won't feel silly anymore I'm just gonna put mathematics in my mind, only mathematics. @Did – Aaron Martinez Apr 26 '17 at 23:38

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