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:
“My facts shall be falsehoods to the common sense. I would so state facts that they shall be significant, shall be myths or mythologic. Facts which the mind perceived, thoughts which the body thought—with these I deal.”
—Henry David Thoreau (1817–1862)
“Science is facts. Just as houses are made of stones, so is science made of facts. But a pile of stones is not a house and a collection of facts is not necessarily science.”
—Jules Henri Poincare (1854–1912)
“Now, what I want is, Facts. Teach these boys and girls nothing but Facts. Facts alone are wanted in life. Plant nothing else, and root out everything else. You can only form the minds of reasoning animals upon Facts: nothing else will ever be of any service to them. This is the principle on which I bring up my own children, and this is the principle on which I bring up these children. Stick to Facts, sir!”
—Charles Dickens (1812–1870)