# Succinct description of situations where naively obvious is correct, but for far more complicated reasons?

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.

• 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. – Bill Dubuque Sep 25 '18 at 14:38
• Mathematicians use the word "obvious" in a bizarre way. I'll fix the wording in a bit. – john mangual Sep 25 '18 at 14:41
• I think the situation is sometimes referred to as an “exercise left for the reader.” – user52817 Sep 25 '18 at 17:12
• @paul So the question is about things that are "more subtle than they appear at first glance", or something more specific? – Bill Dubuque Sep 26 '18 at 22:23
• perhaps it's not even a complete thought yet. I thought i had a question – john mangual Sep 27 '18 at 1:56