Applications of $\mathbb{Q}(\sqrt{5})$ I am willing to give a general audience lecture about prime factorization, and opening towards the lack of unique factorization in the case of e.g. $\mathbb{Q}(\sqrt{5})$. However, I have two issues :

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*how to introduce $\mathbb{Q}(\sqrt{5})$ naturally (for general audience or high-schoolers)? They don't have naturally complex numbers, but maybe I can stay murky about it, or say it is the least we can do to solve $x^2-5=0$ (why is this equation important, though?)

*are there nice, yet accessible, applications (e.g. geometrically, or in cryptography, etc.) of the unique factorization in such fields?

 A: This is how I would do it.

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*Explain what it meant to be prime or irreducible.


*Tell them, preferably with a justification, that there are numbers, such as $\sqrt2, \sqrt3, \cdots$, that are not rational. But be cautious not to go beyond positive square roots.


*That gives you a way to expand the horizon by adding these extra numbers, one at a time. Like $\mathbb{Z}[\sqrt2], \mathbb{Q}[\sqrt2],$
Also, $\mathbb{Z}[\sqrt6]\subset\mathbb{Z}[\sqrt2][\sqrt3]$ while $\mathbb{Q}[\sqrt6]=\mathbb{Q}[\sqrt2][\sqrt3]$


*Tell them, while this gives you some extra freedom (I would show them that $\mathbb{Q}[\Delta]$ is where the solutions of a rational quadratic equation with discriminant $\Delta$ lives or casually mention something like this), sometimes you lose some nice algebraic properties. For example $\mathbb{Z}[\sqrt5]$ is not a unique factorization domain. $$(3-\sqrt5)(3+\sqrt5)=2^2$$
I wouldn't talk about complex numbers or Gaussian integers at all. Later when they see complex numbers, they will remember your talk.
A: Why are you choosing this specific topic of $\mathbf Q(\sqrt{5})$, which really sounds like it is about $\mathbf Z[\sqrt{5}]$?  Students are going to find the notation intimidating.
And is the audience high school students or a true general audience?  If it's a general audience, then your topic is too sophisticated.  Find something simpler, like puzzles (15-puzzle, Rubik's cube).
If the audience is high school students who have already taken algebra and trigonometry, and if the goal is to discuss a setting where there is failure of unique factorization, then use an example they already know but had not thought much about: the system of trigonometric polynomials, which are things like the function
$$
\sin^2(x)\cos^3(x) - 4\sin(x)\cos^2(x) + 2\cos(x) - 9.
$$
It's a polynomial in $\sin x$ and $\cos x$.  You can add and multiply such functions, and thanks to $\sin^2(x) + \cos^2(x) = 1$ you can write the same expression in multiple ways.  In fancier language, these are precisely the finite Fourier series (linear combinations of $\sin(mx)\cos(nx)$ for $m, n \geq 0$), but you don't have to mention that.
Anyway, it turns out that $\sin x$ and $\cos x$ are irreducible: they're not the product of two other nonconstant trigonometric polynomials. And also $1 \pm \sin x$ and $1 \pm \cos x$ are irreducible.  This implies that the familiar relation from trigonometry
$$
\sin^2 x = 1 - \cos^2 x = (1+\cos x)(1 - \cos x)
$$
is an example of nonunique irreducible factorization: $1 + \cos x$ and $1 - \cos x$ are not scale multiples of $\sin x$ (why? neither of the functions $1 + \cos x$ and $1 - \cos x$ vanish at all the values where $\sin x = 0$, so $1\pm \cos x$ isn't a scalar multiple of $\sin x$).  Here is Hale Trotter's short paper about this example: http://alpha.math.uga.edu/~pete/trotter.pdf.
