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:
“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)
“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)
“O for a man who is a man, and, as my neighbor says, has a bone in his back which you cannot pass your hand through! Our statistics are at fault: the population has been returned too large. How many men are there to a square thousand miles in this country? Hardly one.”
—Henry David Thoreau (1817–1862)
“Look at this poet William Carlos Williams: he is primitive and native, and his roots are in raw forest and violent places; he is word-sick and place-crazy. He admires strength, but for what? Violence! This is the cult of the frontier mind.”
—Edward Dahlberg (1900–1977)
“Some people are under the impression that all that is required to make a good fisherman is the ability to tell lies easily and without blushing; but this is a mistake. Mere bald fabrication is useless; the veriest tyro can manage that. It is in the circumstantial detail, the embellishing touches of probability, the general air of scrupulous—almost of pedantic—veracity, that the experienced angler is seen.”
—Jerome K. Jerome (1859–1927)
“‘She has got rings on every finger,
Round one of them she have got three.
She have gold enough around her middle
To buy Northumberland that belongs to thee.”
—Unknown. Young Beichan (l. 61–64)