algebraically closed body (english) - meaning, definition, translation, pronunciation

Part of Speech

Noun Phrase

Phonetic Transcription

/ælˈdʒɛbrəˌkli kloʊzd boʊdi/

Meaning and Usage

The term "algebraically closed body" refers to a mathematical structure, specifically a field, in which every non-constant polynomial equation has a root within that field. This concept is fundamental in abstract algebra and algebraic geometry, as it ensures that certain fundamental properties of polynomials hold true.

The usage in English is primarily within mathematical discourse, particularly in abstract algebra and advanced mathematics textbooks. Its frequency is relatively low compared to everyday language, and it is predominantly used in written context.

Example Sentences

1. The field of complex numbers is an example of an algebraically closed body.

2. Поле комплексных чисел является примером алгебраически замкнутого тела.

3. In calculus, we often encounter algebraically closed bodies when solving polynomial equations.

4. В исчислении мы часто сталкиваемся с алгебраически замкнутыми телами при решении полиномиальных уравнений.

5. Understanding the properties of variable expressions in an algebraically closed body can simplify many mathematical problems.

6. Понимание свойств переменных в алгебраически замкнутом теле может упростить многие математические задачи.

Idiomatic Expressions

The phrase "algebraically closed body" is quite technical and doesn’t feature prominently in idiomatic expressions or colloquial speech. However, here are some examples of sentences that include mathematical idioms:

1. To explore complex problems, one must first ensure the underlying body is algebraically closed.

2. Чтобы исследовать сложные проблемы, сначала нужно убедиться, что основное тело алгебраически замкнуто.

3. When solving quadratic equations, you will find that an algebraically closed body is indispensable.

4. При решении квадратных уравнений вы обнаружите, что алгебраически замкнутое тело незаменимо.

5. The theorem states that in an algebraically closed body, a polynomial of degree n has precisely n roots.

6. Теорема гласит, что в алгебраически замкнутом теле полином степени n имеет ровно n корней.

Etymology

The term "algebraically closed body" stems from the field of algebra, specifically from the Latin word "algebra," which itself originates from the Arabic "al-jabr," meaning "the reunion of broken parts." The term "closed" relates to the closure property in mathematics, indicating that certain algebraic operations result in elements that remain within the defined set. The use of "body" in this context reflects a mathematical structure, typically referred to as a field.

Synonyms and Antonyms

Synonyms: - Algebraically closed field - Algebraically complete field

Antonyms: - Algebraically open field - Non-closed field

This term is mostly used in the context of advanced mathematical study and is less likely to have direct colloquial synonyms or antonyms.