There is no minimal element of . Take any . Then the set does not have a minimal element.I know why a minimal element of I exists, but why can we conclude a minimal positive element exists?
To prove that there is a minimal positive element, it is sufficient to know that there is any positive element. This is natural numbers we are talking about
Take any (w.l.o.g. positive) and the minimal positive element of . Then the GCD of and is a linear combination of and by Bézout's identity, an application of the Euclid's algorithm. From there it is easy to show that divides .Also, how would I apply Euclidean algorithm to something this abstract?