Uniqueness of Square Roots in General Rings
In a ring we call an element b a square root of a iff b2 = a.
In an integral domain, suppose the element a has some square root b, so b2 = a. Then this square root is not necessarily unique, but it is "almost unique" in the following sense: If x too is a square root of a, then x2 = a = b2. So x2 – b2 = 0, or, by commutativity, (x + b)(x – b) = 0. Because there are no zero divisors in the integral domain, we conclude that one factor is zero, and x = ±b. The square root of a, if it exists, is therefore unique up to a sign, in integral domains.
To see that the square root need not be unique up to sign in a general ring, consider the ring from modular arithmetic. Here, the element 1 has four distinct square roots, namely ±1 and ±3. On the other hand, the element 2 has no square root. See also the article quadratic residue for details.
Another example is provided by the quaternions in which the element −1 has an infinitude of square roots including ±i, ±j, and ±k.
In fact, the set of square roots of −1 is exactly
Hence this set is exactly the same size and shape as the (surface of the) unit sphere in 3-space.
Read more about this topic: Sqrt
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