Shouldn't the definition of a prime number be changed to account for negative factors?

At the moment, from what I can gather the current definition of a Prime Number is; "a number that is divisible only by itself and $1$ (e.g. $2, 3, 5, 7, 11$)". However such a prime number like $7$ can also be made by multiplying $-1$ and $-7$. Hence shouldn't the definition be changed to "it can only be divisible by itself and one as well as $-1$ and its negative counterpart"?

This is a good question, and I think it motivates the more general definition of a prime number. Let's begin with units.

A unit is an integer such that it has an integer inverse.

In other words, a unit is an integer $k$ such that there is an integer $l$, with $k\cdot l=1$. What are the units? Well, it is obvious that $1$ is a unit, but $-1$ is also a unit, since $-1\cdot-1=1$. However, there are no other units (can you prove why)?

Now, to prime numbers:

An integer $k$ is prime if it is not a unit, and for any integers $a,b$ such that $a\cdot b=k$, either $a$ or $b$ is a unit.

In other words, a prime number is one that can't be a product of two non-unit integers.

So what are the primes? Well, all of the primes you know ($2,3,5...$) are primes, but so are $-2,-3,-5,...$

The wonderful part about these definitions is that they generalize nicely to more interesting sets of numbers.

P.S. If you learn abstract algebra, you'll see that I've kinda lied to you. The definition of prime numbers I've given you is actually that of irreducible numbers, but for the integers, they are one and the same.

Wikipedia says that a prime is not a product of other natural numbers:

A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers.