Algebraic Integer - Examples

Examples

  • The only algebraic integers which are found in the set of rational numbers are the integers. In other words, the intersection of Q and A is exactly Z. The rational number a/b is not an algebraic integer unless b divides a. Note that the leading coefficient of the polynomial bx − a is the integer b. As another special case, the square root √n of a non-negative integer n is an algebraic integer, and so is irrational unless n is a perfect square.
  • If d is a square free integer then the extension K = Q(√d) is a quadratic field of rational numbers. The ring of algebraic integers OK contains √d since this is a root of the monic polynomial x2 − d. Moreover, if d ≡ 1 (mod 4) the element (1 + √d)/2 is also an algebraic integer. It satisfies the polynomial x2 − x + (1 − d)/4 where the constant term (1 − d)/4 is an integer. The full ring of integers is generated by √d or (1 + √d)/2 respectively. See quadratic integers for more.
  • The ring of integers of the field has the following integral basis, writing for two square-free coprime integers h and k:
\begin{cases} 1, \alpha, \frac{\alpha^2 \pm k^2 \alpha + k^2}{3k} & m \equiv \pm 1 \mod 9 \\ 1, \alpha, \frac{\alpha^2}k & else \end{cases}
  • If ζn is a primitive n-th root of unity, then the ring of integers of the cyclotomic field Q(ζn) is precisely Z.
  • If α is an algebraic integer then is another algebraic integer. A polynomial for β is obtained by substituting xn in the polynomial for α.

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