Facts
- The sum, difference and product of two algebraic integers is an algebraic integer. In general their quotient is not. The monic polynomial involved is generally of higher degree than those of the original algebraic integers, and can be found by taking resultants and factoring. For example, if x2 − x − 1 = 0, y3 − y − 1 = 0 and z = xy, then eliminating x and y from z − xy and the polynomials satisfied by x and y using the resultant gives z6 − 3z4 − 4z3 + z2 + z − 1, which is irreducible, and is the monic polynomial satisfied by the product. (To see that the xy is a root of the x-resultant of z − xy and x2 − x − 1, one might use the fact that the resultant is contained in the ideal generated by its two input polynomials.)
- Any number constructible out of the integers with roots, addition, and multiplication is therefore an algebraic integer; but not all algebraic integers are so constructible: in a naïve sense, most roots of irreducible quintics are not. This is the Abel-Ruffini theorem.
- Every root of a monic polynomial whose coefficients are algebraic integers is itself an algebraic integer. In other words, the algebraic integers form a ring which is integrally closed in any of its extensions.
- The ring of algebraic integers A is a Bézout domain.
Read more about this topic: Algebraic Integer
Famous quotes containing the word facts:
“It is not an arbitrary “decree of God,” but in the nature of man, that a veil shuts down on the facts of to-morrow; for the soul will not have us read any other cipher than that of cause and effect. By this veil, which curtains events, it instructs the children of men to live in to-day.”
—Ralph Waldo Emerson (1803–1882)
“A judge is not supposed to know anything about the facts of life until they have been presented in evidence and explained to him at least three times.”
—Parker, Lord Chief Justice (1900–1972)
“I went to the woods because I wished to live deliberately, to front only the essential facts of life, and see if I could not learn what it had to teach, and not, when I came to die, discover that I had not lived.... I wanted to live deep and suck out all the marrow of life, to live so sturdily and Spartan-like as to put to rout all that was not life, to cut a broad swath and shave close, to drive life into a corner, and reduce it to its lowest terms.”
—Henry David Thoreau (1817–1862)