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This is a nicely paced 13 minute presentation. It goes slowly and carefully over root two, before going faster over root N, then finishing off rapidly with monic polynomials. Suitable for a variety of audiences. But I still cannot figure out how it avoids fancy properties of the integers. I expect that you must at least use Euclid's Algorithm, but it doesn't. Here is how I prove that square roots are either integers or irrationals:

Suppose that √n∈ℚ. Write √n = a/b where a and b are coprime. Then b√n=a. Also, by Bezout's Theorem there are r,s∈ℤ such that ra+sb=1. Multiply by √n

ra√n + sb√n = √n

We already have b√n=a so sb√n is an integer. Does anything similar happen for ra√n ? Yes, just multiply b√n=a by √n obtaining bn=a√n, from which we deduce that ra√n must also be an integer. Thus √n is the sum of two integers and itself an integer.

That is awkward; a rational square root always turns out to be a whole number! Since 1<2<4 we know that 1<√2<2. But there is no integer between one and two so √2 is irrational.

But the approach in the video has neither Euclid's Algorithm, nor Bezout's Theorem (which I view as a corrollary of Euclid's Algorithm).