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root

This entry is about the notion of root in algebra. For the notion in representation theory see at root (in representation theory).

Contents

Definitions

A square root of an element aa in a monoid MM is a solution to the equation x 2=ax^2 = a within MM. If MM is the multiplicative monoid of non-zero elements of a (commutative) field or integral domain KK (or a submonoid of this, such as the group of units), then there are exactly two square roots, denoted ±a\pm \sqrt{a}, if there are any; the element 00 has only one square root, ±0=0\pm \sqrt{0} = 0. But if KK is even a non-integral domain or a non-commutative skew field, then there may be more; in the skew field \mathbb{H} of quaternions, there are continuumly many square roots of 1-1.

More generally, for nn \in \mathbb{N} a solution to the equation x n=ax^n = a is called an nnth root of aa. Specifically for a=1a = 1 and working in the field of complex numbers, one speaks of an nnth 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 3 rd3^{rd}) roots of unity.

More generally, for any polynomial P(x)P(x) of xx with coefficients in a field KK, a solution to P(x)=0P(x) = 0 in KK is called a root of PP. When P(x)K[x]P(x) \in K[x] has no solution in kk, one can speak of a splitting field? obtained by “adjoining roots” of PP to KK, meaning that one considers roots in an extension field? i:KEi: K \hookrightarrow E of the corresponding polynomial Q=(i KK[x])(P)E[x]Q = (i \otimes_K K[x])(P) \in E[x], i.e., applying the evident composite map

K[x]K KK[x]i K1E KK[x]E[x]K[x] \cong K \otimes_K K[x] \stackrel{i \otimes_K 1}{\to} E \otimes_K K[x] \cong E[x]

to PP to get QQ, and passing the smallest intermediate subfield? between KK and EE that contains the designated roots of QQ (often writing PP for QQ by abuse of language).

More generally still, one may refer to roots even of non-polynomial functions ff defined on a field, for example of meromorphic functions f:f \colon \mathbb{C} \to \mathbb{C}, although it is much more usual to speak of zeroes of ff instead of roots of ff (e.g., zeroes of the Riemann zeta function); see zero set? and intermediate value theorem.

Roots of unity in fields

In a field kk, a torsion element of the multiplicative group k *k^\ast is a root of unity by definition. Moreover we have the following useful result.

Theorem

Let GG be a finite subgroup of the multiplicative group k *k^\ast of a field kk. Then GG is cyclic.

Proof

Let ee be the exponent of GG, i.e., the smallest n>0n \gt 0 such that g n=1g^n = 1 for all gGg \in G, and let m=order(G)m = order(G). Then each element of GG is a root of x e1x^e - 1, so that gG(xg)\prod_{g \in G} (x - g) divides x e1x^e - 1, i.e., mem \leq e. But of course g m=1g^m = 1 for all gGg \in G, so eme \leq m, and thus e=me = m.

This is enough to force GG to be cyclic. Indeed, write e=p 1 r 1p 2 r 2p k r ke = p_1^{r_1} p_2^{r_2} \ldots p_k^{r_k}. Since ee is the least common multiple of the orders of elements, the exponent r ir_i is the maximum multiplicity of p ip_i occurring in orders of elements; any element realizing this maximum will have order divisible by p i r ip_i^{r_i}, and some power y iy_i of that element will have order exactly p i r ip_i^{r_i}. Then y= iy iy = \prod_i y_i will have order e=me = m by the following lemma and induction, so that powers of yy exhaust all mm elements of GG, i.e., yy generates GG as desired.

Lemma

If m,nm, n are relatively prime and xx has order mm and yy has order nn in an abelian group, then xyx y has order mnm n.

Proof

Suppose (xy) k=x ky k=1(x y)^k = x^k y^k = 1. For some a,ba, b we have ambn=1a m - b n = 1, and so 1=x kamy kam=y kam=y ky kbn=y k1 = x^{k a m} y^{k a m} = y^{k a m} = y^k y^{k b n} = y^k. It follows that nn divides kk. Similarly mm divides kk, so mn=lcm(m,n)m n = lcm(m, n) divides kk, as desired.

Clearly there is at most one subgroup GG of a given order nn in k *k^\ast, which will be the set of n thn^{th} roots of unity. If GG is a finite subgroup of order nn in k *k^\ast, then a generator of GG is called a primitive n thn^{th} root of unity in kk.

Corollary

Every finite field has a cyclic multiplicative group.

Revised on April 19, 2014 11:03:38 by Todd Trimble (67.81.95.215)