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: Square Root
Famous quotes containing the words uniqueness of, uniqueness, square, roots, general and/or rings:
“Until now when we have started to talk about the uniqueness of America we have almost always ended by comparing ourselves to Europe. Toward her we have felt all the attraction and repulsions of Oedipus.”
—Daniel J. Boorstin (b. 1914)
“Somehow we have been taught to believe that the experiences of girls and women are not important in the study and understanding of human behavior. If we know men, then we know all of humankind. These prevalent cultural attitudes totally deny the uniqueness of the female experience, limiting the development of girls and women and depriving a needy world of the gifts, talents, and resources our daughters have to offer.”
—Jeanne Elium (20th century)
“If magistrates had true justice, and if physicians had the true art of healing, they would have no occasion for square caps; the majesty of these sciences would of itself be venerable enough. But having only imaginary knowledge, they must employ those silly tools that strike the imagination with which they have to deal; and thereby, in fact, they inspire respect.”
—Blaise Pascal (16231662)
“He who sins easily, sins less. The very power
Renders less vigorous the roots of evil.”
—Ovid (Publius Ovidius Naso)
“I never saw any people who appeared to live so much without amusement as the Cincinnatians.... Were it not for the churches,... I think there might be a general bonfire of best bonnets, for I never could discover any other use for them.”
—Frances Trollope (17801863)
“Ah, Christ, I love you rings to the wild sky
And I must think a little of the past:
When I was ten I told a stinking lie
That got a black boy whipped....”
—Allen Tate (18991979)