mathematics

Ring of integers

Ring of integers 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 Ring of integers rather than just read about it. In short: In mathematics, the ring of integers of an algebraic number field K {\displaystyle K} (also sometimes called the number ring corresponding to number field K {\displaystyle K} ) is the ring of all algebraic integers contained in K {\displaystyle K} . An algebraic integer is a root of a monic polynomial with integer coefficients: x n + c n − 1 x n − 1 + ⋯ + c 0 {\displaystyle x^{n}+c_{n-1}x^{n-1}+\cdots +c_{0}} .

Key takeaways

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

Reference excerpt

In mathematics, the ring of integers of an algebraic number field K {\displaystyle K} (also sometimes called the number ring corresponding to number field K {\displaystyle K} ) is the ring of all algebraic integers contained in K {\displaystyle K} . An algebraic integer is a root of a monic polynomial with integer coefficients: x n + c n − 1 x n − 1 + ⋯ + c 0 {\displaystyle x^{n}+c_{n-1}x^{n-1}+\cdots +c_{0}} . This ring is often denoted by O K {\displaystyle O_{K}} or O K {\displaystyle {\mathcal {O}}_{K}} . Since any integer belongs to K {\displaystyle K} and is an integral element of K {\displaystyle K} , the ring Z {\displaystyle \mathbb {Z} } is always a subring of O K {\displaystyle O_{K}} . The ring of integers Z {\displaystyle \mathbb {Z} } is the simplest possible ring of integers. Namely, Z = O Q {\displaystyle \mathbb {Z} =O_{\mathbb {Q} }} where Q {\displaystyle \mathbb {Q} } is the field of rational numbers. And indeed, in algebraic number theory the elements of Z {\displaystyle \mathbb {Z} } are often called the "rational integers" because of this. The next simplest example is the ring of Gaussian integers Z [ i ] {\displaystyle \mathbb {Z} [i]} , consisting of complex numbers whose real and imaginary parts are integers. It is the ring of integers in the number field Q ( i ) {\displaystyle \mathbb {Q} (i)} of Gaussian rationals, consisting of complex numbers whose real and imaginary parts are rational numbers. Like the rational integers, Z [ i ] {\displaystyle \mathbb {Z} [i]} is a Euclidean domain. The ring of integers of an algebraic number field is the unique maximal order in the field. It is always a Dedekind domain.

Properties The ring of integers OK is a finitely-generated Z {\displaystyle \mathbb {Z} } -module. Indeed, it is a free Z {\displaystyle \mathbb {Z} } -module, and thus has an integral basis, that is a basis b1, ..., bn ∈ OK of the Q {\displaystyle \mathbb {Q} } -vector space K such that each element x in OK can be uniquely represented as

x = ∑ i = 1 n a i b i , {\displaystyle x=\sum _{i=1}^{n}a_{i}b_{i},}

with a i ∈ Z {\displaystyle a_{i}\in \mathbb {Z} } . The rank n of OK as a free Z {\displaystyle \mathbb {Z} } -module is equal to the degree of K over Q {\displaystyle \mathbb {Q} } .

Examples

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with Ring of integers

Start with the simplest possible case. Write down what Ring of integers 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 Ring of integers 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 Ring of integers 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 Ring of integers

In research: Ring of integers 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 Ring of integers 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: Ring of integers is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic number theory, Ring theory, so understanding it makes those chapters shorter.
In everyday life: Look for Ring of integers 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 Ring of integers in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Ring of integers 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 Ring of integers out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Ring of integers in simple terms?

Why does Ring of integers 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 Ring of integers?

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 Ring of integers.