Algebraic integers

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Algebraic integers are algebraic numbers which are roots of irreducible monic polynomials with integer coefficients

$x^{n}+a_{n-1}\,x^{n-1}+\cdots +a_{2}\,x^{2}+a_{1}\,x+a_{0}=0,\,$

where $\scriptstyle a_{n-1},\,\ldots ,\,a_{2},\,a_{1},\,a_{0}\,\in \,\mathbb {Z} .\,$

For example, $\scriptstyle 4+{\sqrt {7}}\,$ and $\scriptstyle {\frac {1+{\sqrt {7}}}{2}}\,$ are algebraic integers. By contrast, $\scriptstyle {\frac {4+{\sqrt {7}}}{2}}\,$ is an algebraic number but not an algebraic integer.

Some computer algebra systems have built-in functions to test whether a given number is an algebraic integer. For example, the function AlgebraicIntegerQ in Wolfram Mathematica. (Warning: this function returns False when it cannot determine the answer.)

Rational integers (algebraic integers of degree 1)

The rational integers are precisely the (ordinary) integers, i.e., members of $\scriptstyle \mathbb {Z} \,$. The minimal polynomial for $\scriptstyle n\in \mathbb {Z}$ is thus simply $x-n$. For example, 7 is an algebraic integer of degree 1, with its minimal polynomial being $x-7$.

Quadratic integers (algebraic integers of degree 2)

The quadratic integers are quadratic numbers which are roots of irreducible monic polynomials with integer coefficients of degree 2.

All Gaussian integers are quadratic integers, as are all Eisenstein integers. Other examples of quadratic integers include $\scriptstyle {\sqrt {2}}$, $\scriptstyle 3+7{\sqrt {-5}}$ (or $\scriptstyle 3+7i{\sqrt {5}}$), and the golden ratio $\scriptstyle {\frac {1+{\sqrt {5}}}{2}}.$

Examples of numbers that are not quadratic integers (although they are quadratic numbers) include $\scriptstyle {\frac {1}{\sqrt {2}}}$ and $\scriptstyle {\frac {1+{\sqrt {12}}}{2}}$.

Cubic integers (algebraic integers of degree 3)

The cubic integers are cubic numbers which are roots of irreducible monic polynomials with integer coefficients of degree 3.

Arithmetic integers

Arithmetic numbers are those which can be expressed with a finite number of algebraic operations, which consist of field operations (+, −, ×, /) and root extraction, hence all arithmetic numbers are algebraic. Conversely, all algebraic numbers (thus all algebraic integers) of degree up to 4 are arithmetic numbers, but not all algebraic numbers of degree 5 and above are arithmetic. For example, the roots of the irreducible monic polynomial $x^{5}+x+1=0$ are not arithmetic. (The roots of the monic polynomials $p(x)=x^{5}+ax^{3}+bx^{2}+cx+d=0$, if $p(x)$ is irreducible [i.e. cannot factored into polynomials, with integer coefficients, of lower degree], and if $a$ and $b$ are even and $c$ and $d$ are odd, are not arithmetic.)