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What is the name for a situation where the obvious thing turns out to be true, but the reasoning is more complicated?

In abstract algebra we can say the rational numbers - the fractions, $\mathbb{Q}$ form a field. Then we can adjoin the number $x = \sqrt{-1}$ which solves $x^2 + 1 = 0$, and we can have the field $\mathbb{Q}(\sqrt{-1}) = \mathbb{Q}[x]/(x^2 + 1)$.

Doesn't it make sense that the ring of integers should be $\mathbb{Z}[i]$. We are going to take the ordinary integers $\mathbb{Z}$ and adjoin the number $i = \sqrt{-1}$. Unfortunately, this conflicts with the definition of ring of integers.

Definition The ring of integers $\mathcal{O}_K$ is the set of solutions $x \in K$ of monic irreducible polynomials $x^n + a_0 x^{n-1} + \dots + a_{n-1}$ with coefficients in $\mathbb{Q}$

And then it's not at all obvious that $\mathcal{O}_{\mathbb{Q}(i)} = \mathbb{Z}[i]$. Would it be safe to skip this definition entirely and just say here's an interesting mathematical object:

$$ \mathbb{Z}[i] = \big\{ a + bi : a , b \in \mathbb{Z} \big\} $$

and discuss whatever topic of interest, e.g. unique factorization.

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    $\begingroup$ Why do you believe it is "obvious"? The naive approach fails in other cases, adjoining $\sqrt{-3}$ to $\Bbb Z$ yields a ring $R = \Bbb Z[\sqrt{-3}]$ that is not integrally closed since the proper fraction $\omega = (-1+\sqrt{-3})/2$ over $R$ is a root of $x^2+x+1$. Thus RRT = Rational Root Test fails, so the ring lacks unique factorization (because UFD $\Rightarrow$ RRT). So we need to enlarge the "naive" ring of integers to be integrally closed if there is any hope that it will enjoy unique factorization. $\endgroup$ – Bill Dubuque Sep 25 '18 at 14:38
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    $\begingroup$ Mathematicians use the word "obvious" in a bizarre way. I'll fix the wording in a bit. $\endgroup$ – john mangual Sep 25 '18 at 14:41
  • $\begingroup$ I think the situation is sometimes referred to as an “exercise left for the reader.” $\endgroup$ – user52817 Sep 25 '18 at 17:12
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    $\begingroup$ @paul So the question is about things that are "more subtle than they appear at first glance", or something more specific? $\endgroup$ – Bill Dubuque Sep 26 '18 at 22:23
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    $\begingroup$ perhaps it's not even a complete thought yet. I thought i had a question $\endgroup$ – john mangual Sep 27 '18 at 1:56

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