mathematics

Lagrange's theorem (number theory)

Lagrange's theorem (number theory) is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Lagrange's theorem (number theory) rather than just read about it. In short: In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime p. More precisely, it states that for all integer polynomials f ∈ Z [ x ] {\displaystyle \textstyle f\in \mathbb {Z} [x]} , either: every coefficient of f is divisible by p, or p ∣ f ( x ) {\displaystyle p\mid f(x)} has at most deg f sol…

Key takeaways

Lagrange's theorem (number theory) belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Lagrange's theorem (number theory) to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Lagrange's theorem (number theory) from memory before moving on to harder problems.

Reference excerpt

In number theory, Lagrange's theorem is a statement named after Joseph-Louis Lagrange about how frequently a polynomial over the integers may evaluate to a multiple of a fixed prime p. More precisely, it states that for all integer polynomials f ∈ Z [ x ] {\displaystyle \textstyle f\in \mathbb {Z} [x]} , either:

every coefficient of f is divisible by p, or

p ∣ f ( x ) {\displaystyle p\mid f(x)} has at most deg f solutions in {1, 2, ..., p}, where deg f is the degree of f. This can be stated with congruence classes as follows: for all polynomials f ∈ ( Z / p Z ) [ x ] {\displaystyle \textstyle f\in (\mathbb {Z} /p\mathbb {Z} )[x]} with p prime, either:

every coefficient of f is null, or

f ( x ) = 0 {\displaystyle f(x)=0} has at most deg f solutions in Z / p Z {\displaystyle \mathbb {Z} /p\mathbb {Z} } . If p is not prime, then there can potentially be more than deg f(x) solutions. Consider for example p=8 and the polynomial f(x)=x2−1, where 1, 3, 5, 7 are all solutions.

Proof Let f ∈ Z [ x ] {\displaystyle \textstyle f\in \mathbb {Z} [x]} be an integer polynomial, and write g ∈ (Z/pZ)[x] the polynomial obtained by taking its coefficients mod p. Then, for all integers x,

f ( x ) ≡ 0 ( mod p ) ⟺ g ( x ) ≡ 0 ( mod p ) {\displaystyle f(x)\equiv 0{\pmod {p}}\quad \Longleftrightarrow \quad g(x)\equiv 0{\pmod {p}}} . Furthermore, by the basic rules of modular arithmetic,

f ( x ) ≡ 0 ( mod p ) ⟺ f ( x mod p ) ≡ 0 ( mod p ) ⟺ g ( x mod p ) ≡ 0 ( mod p ) {\displaystyle f(x)\equiv 0{\pmod {p}}\quad \Longleftrightarrow \quad f(x{\bmod {p}})\equiv 0{\pmod {p}}\quad \Longleftrightarrow \quad g(x{\bmod {p}})\equiv 0{\pmod {p}}} . Both versions of the theorem (over Z and over Z/pZ) are thus equivalent. We prove the second version by induction on the degree, in the case where the coefficients of f are not all null. If deg f = 0 then f has no roots and the statement is true. If deg f ≥ 1 without roots then the statement is also trivially true. Otherwise, deg f ≥ 1 and f has a root k ∈ Z / p Z {\displaystyle k\in \mathbb {Z} /p\mathbb {Z} } . The fact that Z/pZ is a field allows to apply the division algorithm to f and the polynomial x − k (of degree 1), which yields the existence of a polynomial g ∈ ( Z / p Z ) [ x ] {\displaystyle \textstyle g\in (\mathbb {Z} /p\mathbb {Z} )[x]} (of degree lower than that of f) and of a constant r ∈ Z / p Z {\displaystyle \textstyle r\in \mathbb {Z} /p\mathbb {Z} } (of degree lower than 1) such that

f ( x ) = g ( x ) ( x − k ) + r . {\displaystyle f(x)=g(x)(x-k)+r.}

Evaluating at x = k provides r = 0. The other roots of f are then roots of g as well, which by the induction property are at most deg g ≤ deg f − 1 in number. This proves the result.

Generalization Let p(X) be a polynomial over an integral domain R with degree n > 0. Then the polynomial equation p(x) = 0 has at most n = deg(p(X)) roots in R.

References

LeVeque, William J. (2002) [1956]. Topics in Number Theory, Volumes I and II. New York: Dover Publications. p. 42. ISBN 978-0-486-42539-9. Zbl 1009.11001. Tattersall, James J. (2005). Elementary Number Theory in Nine Chapters (2nd ed.). Cambridge University Press. p. 198. ISBN 0-521-85014-2. Zbl 1071.11002.

Worked examples

Example 1 — a first encounter with Lagrange's theorem (number theory)

Start with the simplest possible case. Write down what Lagrange's theorem (number theory) claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Lagrange's theorem (number theory) before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Lagrange's theorem (number theory) ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Lagrange's theorem (number theory)

In research: Lagrange's theorem (number theory) appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Lagrange's theorem (number theory) in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Lagrange's theorem (number theory) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Theorems about polynomials, Theorems about prime numbers, so understanding it makes those chapters shorter.
In everyday life: Look for Lagrange's theorem (number theory) outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Lagrange's theorem (number theory) in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Lagrange's theorem (number theory) means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Lagrange's theorem (number theory) out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Lagrange's theorem (number theory) in simple terms?

Why does Lagrange's theorem (number theory) matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Lagrange's theorem (number theory)?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Lagrange's theorem (number theory).