number theory Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777� ...

, Lagrange's theorem is a statement named after

Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangiapolynomial In mathematics, a polynomial is an expression consisting of indeterminates (also called variables) and coefficients, that involves only the operations of addition, subtraction, multiplication, and positive-integer powers of variables. An exa ...

over the

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

s may evaluate to a multiple of a fixed

prime A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

. More precisely, it states that if ''p'' is a prime number,

x \in \mathbb/p\mathbb

, and

\textstyle f(x) \in \mathbb /math> is a polynomial with integer coefficients, then either:
* every coefficient of  is divisible by ''p'', or
*  has at most  solutions

where  is the

degree Degree may refer to: As a unit of measurement * Degree (angle), a unit of angle measurement ** Degree of geographical latitude ** Degree of geographical longitude * Degree symbol (°), a notation used in science, engineering, and mathematics ...

of . If the modulus is not prime, then it is possible for there to be more than solutions.

A proof of Lagrange's theorem

The two key ideas are the following. Let be the polynomial obtained from by taking the coefficients . Now: # is divisible by if and only if ; and # has no more than roots. More rigorously, start by noting that if and only if each coefficient of is divisible by . Assume ; its degree is thus well-defined. It is easy to see . To prove (1), first note that we can compute either directly, i.e. by plugging in (the

residue class In mathematics, modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" when reaching a certain value, called the modulus. The modern approach to modular arithmetic was developed by Carl Friedrich Gauss in his bo ...

of) and performing arithmetic in , or by reducing . Hence if and only if , i.e. if and only if is divisible by . To prove (2), note that is a

field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...

, which is a standard fact (a quick proof is to note that since is prime, is a finite

integral domain In mathematics, specifically abstract algebra, an integral domain is a nonzero commutative ring in which the product of any two nonzero elements is nonzero. Integral domains are generalizations of the ring of integers and provide a natural set ...

, hence is a field). Another standard fact is that a non-zero polynomial over a field has at most as many roots as its degree; this follows from the

division algorithm A division algorithm is an algorithm which, given two integers N and D, computes their quotient and/or remainder, the result of Euclidean division. Some are applied by hand, while others are employed by digital circuit designs and software. Divis ...

. Finally, note that two solutions are incongruent if and only if

k_1 \not\equiv k_2

. Putting everything together, the number of incongruent solutions by (1) is the same as the number of roots of , which by (2) is at most , which is at most .

References

* * {{DEFAULTSORT:Lagrange's Theorem (Number Theory) Theorems about prime numbers Theorems about polynomials