1 is neither prime nor composite, it is a unit, and I think you need to be clear that “its own self” should be a number other than −1 or 1.
I wanted to say that to you before someone else came along and put it a lot less politely. Or gave a stupid reason, like the fundamental theorem of arithmetic.
Multiplying an algebraic number by a unit does not change the absolute value of the integer term of its polynomial. For example, 3 + √2 is an algebraic integer with minimal polynomial x² − 6x + 7, and 1 + √2 is a unit just like 1.
Then, 3 + √2 multiplied by 1 + √2 is 5 + 4√2, which has minimal polynomial x² − 10x − 7. So −7 instead of 7, but the absolute value is still 7. Multiplying by 1 does not change the number at all, so the minimal polynomial remains exactly the same.
Multiplying an algebraic integer by a prime does change the absolute value of the integer term of the polynomial, and so does multiplying by a composite number.
The Mersenne primes are primes in Z, but, except for 3, they’re all composite in Z[√2].
- (3 − √2)(3 + √2) = 7
- (7 − 3√2)(7 + 3√2) = 31
- (15 − 7√2)(15 + 7√2) = 127
I leave it to someone else to work out the Z[√2] factors of the most recently discovered Mersenne prime.