Are primes extendable to the reals?

It seems like there should be real numbers that do not have factors other than themselves and one. My reasoning:
if x * y = z,
Then we can have an ė, (x + kė) * (y + kė) = z + ė, where k > 0, k < 1, ė > 0
However, as ė->0, then kė vanishes slower, so z + ė should increase, but it can't, because ė is vanishing. This, there must be real numbers that don't have other factors.

Fields have no (proper) ideals.

every real number can be divided by 2

EXPAND MY ANUS

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Any field which is an extension of the rationals does not conform to the idea that there are numbers having only themselves and one as factors. Proof:

Given a field F, which is an extension of, or itself, the rationals, and a number x in F.
2 is in F, x is in F, thus x/2 is in F.
Thus x can be factorised into (x/2)*2
QED