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This is admittedly a bit of a broad question I realize, but curiosity has struck me a bit lately. So before you tl;dr; here's the central question: Are there any known generalizable results that arise when creating towers of integral extensions by adjoining repeated square roots of $i$? Such results might include things like always being a Euclidean Domain in the same way that the Gaussian Integers are. Is there a natural choice of "norm" in each successive integral extension? Is there any recommended material on this subject? I had trouble coming up with search terms on here and on other search engines trying to research this.

I've been studying the properties of the Gaussian Integers, and got to wondering what happens when we adjoin a square root of $i$ (or $\alpha$ if you'd rather not be painted into visualizing them in the complex plane, so long as $\alpha^2 = -1$). In other words, I think it's clear that $x^2 - \alpha$ is irreducible in the ring of polynomials with coefficients in $\mathbb{Z}[\alpha]$. ( $\mathbb{Z[\alpha]}[X]$ ).

Suppose we form an integral extension of $\mathbb{Z}[\alpha]$ by adjoining $\beta$ such that $\beta^2 = \alpha$. Thus, $x^2 - \alpha$ can factor as $(x + \beta)(x - \beta)$. I realize that we can alternatively adjoin a root of $x^2 + \alpha$, but I would imagine that the result would be the same. Specifically, $x^2 + \alpha$ could factor as $$(x + \alpha\beta)(x - \alpha\beta) = x^2 - {\alpha}^2{\beta}^2 = x^2 + \beta^2 = x^2 + \alpha$$ It also occurs to me that one could start at $\mathbb{Z}$ and extend using the irreducible $x^4 + 1$ but I do not have a firm enough grasp of this topic to know whether the resulting extension ring would be isomorphic to $\mathbb{Z}[\alpha, \beta]$.

After some time I convinced myself that all elements in $\mathbb{Z}[\alpha, \beta]$ could be expressed as

$$y = y_0 + y_1\alpha + y_2\beta + y_3\gamma\quad\quad~y_k \in \mathbb{Z}$$

where $\gamma$ denotes $ab$. Originally I did not consider the $\gamma$ portion relevant, but when I drew up some graphs it seemed apparent that each of these four elements represent positions on four independent axes or "number-lines". That is, -1 and 1 are units on the same number-line, but $\beta$ and $\gamma$ certainly are not. Also by "independent", here I don't mean linearly independent in the Linear Algebra sense, but more in the sense that for a unit in the ring to transfer to another "axis" one must multiply it by a unit of the ring other then 1 or -1. Or what many would just intuit as a rotation in the complex plane. And of course it lends itself to a four dimensional module / space of some sort too. At any rate it suggests that the dimension of each successive extension of this type (e.g. $\mathbb{Z}[\alpha, \beta, \sqrt{\beta}]$) would be twice that of the lower ring.

I also observed that this ring has an element that functions as $\sqrt{2}$:

$$s = (\gamma - \beta) $$ $$s^2 = \gamma^2 - 2\beta\gamma + \beta^2$$ $$ = -\alpha -2(-1) + \alpha = 2$$

Which... makes things a bit more interesting than I first expected to see. It made me wonder if this ring is "dense" within another ring, as its lattice points seem to be capable of approximating an arbitrary complex number but I could just be jumping to conclusions there.

I wrote a script to determine whether any of the rational primes that didn't split in $\mathbb{Z}[\alpha]$ would split in this integral extension. For now it's just a crude script but I did manage to find what appear to be nontrivial factorizations of rational primes that are also Gaussian primes ({3, 7, 11, ...}). I was not expecting this to happen, so it was a bit of a pleasant surprise. I haven't yet run the analysis to determine if the factorizations that I found are all equivalent up to units.

An example of what appears to be a nontrivial factorization:

$$ 7 = (-2 - 2\alpha + \beta)(-2 - 2\alpha + \gamma) $$

Once I saw that rational primes $p \equiv 3 \mod 4$ could split, I wondered if I could construct a norm. The best I have come up with that made sense so far was $$N(y) = (y_0^2 + y_1^2 + y_2^2 + y_3^2)^2 - 2(y_0y_2 - y_0y_3 + y_1y_2 + y_1y_3)^2$$

The above evaluated on 7 and its factors: $$N(7) = 7^4$$ $$N(-2 - 2\alpha + \beta + 0\gamma) = (9)^2 - 2(-2 - 2)^2 = 7^2$$ $$N(-2 - 2\alpha + 0\beta + \gamma) = (9)^2 - 2(-2 - 2)^2 = 7^2$$

The above "norm" was constructed by "cheating" a little and designating $\beta = \frac{1}{\sqrt{2}} + \frac{\alpha}{\sqrt{2}}$ to separate into real and imaginary parts, taking the Gaussian Integer norm of that which results in a number of the form $z_0 + z_1\sqrt{2}$, $ z_k \in \mathbb{Z}$. From there we rationalize (well, integer-ize) by multiplying by $z_0 - z_1\sqrt{2}$, which is what $N(y)$ above works out to.

On the other hand this does give rise to things that don't make a whole lot of sense to me. Namely, suppose this is a norm such that $N(y) = 1$ implies that $y$ is unit. Then it would appear that we have more units than one would expect, namely, $-2 - 2\alpha + 3\beta$. I don't yet know if this element in particular is a unit, but if it is I suspect that there are more surprise units lurking out there, or at least, more elements $y$ with $N(y) = 1$ (that are not associates of the preceding number).

Any suggestions on literature and names of topics to look up would be much appreciated. If you scrolled down too far and missed the core question it's at the top in bold like this.

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  • $\begingroup$ Sorry, I couldn’t bring myself to read the whole thing, but your topic is just a fractional part of the study of Cyclotomic Fields. There is a lot to say, and the whole thing is far more than a curiosity in number theory. If I had world enough and time tonight, I’d tell you just a bit of the story. $\endgroup$ – Lubin Feb 3 '18 at 3:27
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Your question presents at least 2 aspects :

1) The Galois theoretic aspect : Let $p$ be a prime number and $q=p$ if $p$ is odd, $4$ if $p=2$. For any $r \ge 1$, let $\zeta_r$ be a primitive $r$-th root of unity. It is classically known that for any $n \ge 1$, the cyclotomic extension $\mathbf Q(\zeta_{q{p}^n})/\mathbf Q$ is normal, with Galois group $G_n\cong ((\mathbf Z/ {q{p}^n}\mathbf Z)^*, \times) \cong (\mathbf Z/q \mathbf Z)^*, \times) \times (\mathbf Z/p^n , +)$. The union of these cyclotomic extensions, denoted $\mathbf Q(\zeta_{{p}^\infty})$, is an infinite Galois extension of $\mathbf Q$, with Galois group $G_{\infty} \cong ((\mathbf Z/q \mathbf Z)^*, \times) \times (\mathbf Z_p , +)$, where $\mathbf Z_p$ denotes the ring of $p$-adic integers. So on the Galois side, everything is known. Of importance is the so called cyclomic $\mathbf Z_p$-extension $\mathbf Q_{\infty} /\mathbf Q$, defined as the union of the extensions $\mathbf Q_{(n)} /\mathbf Q$, where $\mathbf Q_{(n)}$ is the subfield of $\mathbf Q(\zeta_{q{p}^n})$ fixed by $(\mathbf Z/q \mathbf Z)^*$. It allows to define, for any number field $K$, the cyclotomic $\mathbf Z_p$-extension of $K$ by $K_{\infty} = K.\mathbf Q_{\infty}$. More generally, an infinite Galois extension $K_{\infty} /K=\cup K_n$ with Galois group $\Gamma \cong (\mathbf Z_p , +)$ is called a $\mathbf Z_p$-extension of $K$.

2) On the arithmetic side : The ring of integers of $\mathbf Q(\zeta_{q{p}^n})$ is $\mathbf Z(\zeta_{q{p}^n})$, but in general you cannot expect any further property (principality or euclidianity would be a naive hope). So much the better, actually. The algebraic part of Iwasawa theory studies the asymptotic behaviour of the $p$-class groups $A_n$ of the $K_n$ 's when going up to $K_{\infty}$ (naturally, $p$ plays a special role). One emblematic result is Iwasawa's formula giving the $p$-adic valuation of the order of $A_n$, which is asymptotically of the form $\mu p^n + \lambda n + \nu$, where $\mu, \lambda \in \mathbf N$ and $\nu \in \mathbf Z$ are constants depending only on $K_{\infty} /K$. There are deep conjectures (as yet unsolved in general) on the possible vanishing of $\mu, \lambda$ when $K_{\infty}$ is the cyclotomic $\mathbf Z_p$-extension of $K$.

3) When $K_{\infty}$ is the cyclotomic $\mathbf Z_p$-extension of a totally real number field $K$, a third, much deeper aspect of Iwasawa theory is the profoundly mysterious (yet natural) connection that it establishes between the $\mathbf Z_p [[\Gamma]]$-structure of the projective limit of the $p$-class groups and the analytic $p$-adic $L$-functions attached to $K$ (these interpolate the special values of the classical complex $L$-functions). This was the so called Main Conjecture, now a theorem of Mazur and Wiles. Of course the theory does not stop there, but things get more and more elaborate and hard to explain. A good introduction at the graduate level is Washington's book "Introduction to Cyclotomic Fields" .

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  • $\begingroup$ @ngyuyen-quang-do Thank you. I will be having a look at that book. $\endgroup$ – mathonnapkins Feb 7 '18 at 14:59

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