symmetric monoidal (∞,1)-category of spectra
A square root of an element in a monoid is a solution to the equation within . If is the multiplicative monoid of non-zero elements of a (commutative) field or integral domain (or a submonoid of this, such as the group of units), then there are exactly two square roots, denoted , if there are any; the element has only one square root, . But if is even a non-integral domain or a non-commutative skew field, then there may be more; in the skew field of quaternions, there are continuumly many square roots of .
More generally, for a solution to the equation is called an th root of . Specifically for and working in the field of complex numbers, one speaks of an th root of unity. This terminology can be applied to other fields as well; for example, the field of 7-adic numbers contains non-trivial cube (or ) roots of unity.
More generally, for any polynomial of with coefficients in a field , a solution to in is called a root of . When has no solution in , one can speak of a splitting field? obtained by “adjoining roots” of to , meaning that one considers roots in an extension field? of the corresponding polynomial , i.e., applying the evident composite map
to to get , and passing the smallest intermediate subfield? between and that contains the designated roots of (often writing for by abuse of language).
More generally still, one may refer to roots even of non-polynomial functions defined on a field, for example of meromorphic functions , although it is much more usual to speak of zeroes of instead of roots of (e.g., zeroes of the Riemann zeta function); see zero set and intermediate value theorem.
Let be the exponent of , i.e., the smallest such that for all , and let . Then each element of is a root of , so that divides , i.e., . But of course for all , so , and thus .
This is enough to force to be cyclic. Indeed, write . Since is the least common multiple of the orders of elements, the exponent is the maximum multiplicity of occurring in orders of elements; any element realizing this maximum will have order divisible by , and some power of that element will have order exactly . Then will have order by the following lemma and induction, so that powers of exhaust all elements of , i.e., generates as desired.
If are relatively prime and has order and has order in an abelian group, then has order .
Suppose . For some we have , and so . It follows that divides . Similarly divides , so divides , as desired.