Fundamental theorem of algebra: RMO training

It is quite well-known that any positive integer can be factored into a product of primes in a unique way, up to an order. (And, that 1 is neither prime nor composite) —- we all know this from our high school practice of “tree-method” of prime factorization, and related stuff like Sieve of Eratosthenes. But, it is so obvious, and so why it call it a theorem, that too “fundamental” and yet it seems it does not require a proof. It was none other than the prince of mathematicians of yore, Carl Friedrich Gauss, who had written a proof to it. It DOES require a proof — there are some counter example(s). Below is one, which I culled for my students:

Question:

Let $E= \{a+b\sqrt{-5}: a, b \in Z\}$

(a) Show that the sum and product of elements of E are in E.

(b) Define the norm of an element $z \in E$ by $||z||=||a+b\sqrt{-5}||=a^{2}+5b^{2}$. We say that an element $p \in E$ is prime if it is impossible to write $p=n_{1}n_{2}$ with $n_{1}, n_{2} \in E$, and $||n_{1}||>1$, $||n_{2}||>1$; we say that it is composite if it is not prime. Show that in E, 3 is a prime number and 29 is a composite number.

(c) Show that the factorization of 9 in E is not unique.

Cheers,

Nalin Pithwa.

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