Root

An $n$-th root of a number $a$ is a number $x=a^{1/n}$ whose $n$-th power $x^n$ is equal to $a$.

A root of an algebraic equation over a field $k$,

$$a_0x^n+\ldots+a_{n-1}x+a_n=0,$$

is an element $c$ belonging to $k$ or to an extension of $k$ (cf. Extension of a field) such that when $c$ is substituted for $x$ the equation becomes an identity. A root of this equation is also called a root or zero of the polynomial

$$f(x)=a_0x^n+\ldots+a_{n-1}x+a_n.$$

If $c$ is a root of a polynomial $f(x)$, then $f(x)$ is divisible (without remainder) by $x-c$ (see Bezout theorem). Every polynomial $f(x)$ with real or complex coefficients has at least one root (hence as many roots as its degree, counting multiplicities). The polynomial $f(x)$ may be expressed as a product

$$f(x)=a_0(x-c_1)\ldots(x-c_n),$$

where $c_1,\ldots,c_n$ are its roots. If some of the roots $c_1,\ldots,c_n$ of $f(x)$ are equal, their common value is called a multiple root (if a root occurs $m$ times, $m$ is called the multiplicity of that root).

A root of unity is an element of a field $k$ satisfying the equation $x^m=1$ for some natural number $m$. The roots of unity form a subgroup of the multiplicative group of $k$. Conversely, all elements of any finite subgroup of the multiplicative group of a field $k$ are roots of unity (cf. Fermat little theorem) and the subgroup itself is cyclic. This is true, in particular, for the subgroup $U_n$ of all roots of unity of a given degree $n$ contained in the algebraic closure $\bar k$ of $k$, i.e. the subgroup of all $\zeta\in\bar k$ satisfying the equation $\zeta^n=1$. If $n$ is relatively prime to the characteristic of $k$ (or if the characteristic is 0), then the group $U_n$ is of order $n$ and its generators are known as primitive $n$-th roots of unity. The number of such roots in $U_n$ is given by the Euler function $\phi(n)$, i.e. the number of residues $\bmod\,n$ which are relatively prime to $n$. In a field of characteristic $p>0$ there are no $p$-th roots of unity other than 1.

If the field $k$ is finitely generated over its prime subfield, then the number of roots of unity in $k$ is finite.

In the field of complex numbers, a number $z$ is an $n$-th root of unity if and only if $|z|=1$ and $\arg z=2\pi m/n$, where $m$ and $n$ are integers, i.e. if and only if

$$z=e^{2\pi im/n}=\cos\frac{2\pi m}{n}+i\sin\frac{2\pi m}{n};$$

in this case the primitive roots of unity are exactly those for which $(m,n)=1$. In the complex plane, the $n$-th roots of unity coincide with the vertices of the regular $n$-gon inscribed in the unit circle; this explains the connection of roots of unity with the problem of squaring the circle (construction of polygons, cf. Geometric constructions).

Roots of unity appear in number theory as the values of various important number-theoretical functions (Abelian numerical characters; Legendre symbol; Möbius function; norm-residue symbol; etc.). In field theory and algebraic number theory an important position is occupied by fields obtained by adjunction of roots of unity to some ground field (see Cyclotomic field; Cyclotomic extension; Kummer extension).

References

[1]	B.L. van der Waerden, "Algebra" , 1–2 , Springer (1967–1971) (Translated from German)
[2]	S. Lang, "Algebra" , Addison-Wesley (1984)

Comments

For the concept of a root in Lie algebra theory see Lie algebra, semi-simple and Root system.