ultrametric field (english) - meaning, definition, translation, pronunciation

Part of Speech

Noun Phrase

Phonetic Transcription

/ˌʌltrəˈmɛtrɪk fiːld/

Meaning and Usage

An ultrametric field is a concept from mathematics and specifically a branch of metric space theory. It refers to a set equipped with a distance function that satisfies the ultrametric inequality, which is stronger than the usual triangle inequality for metrics. In an ultrametric space, the distance between points satisfies the following condition:
d(x, z) ≤ max{d(x, y), d(y, z)} for any points x, y, and z in the space.

Ultrametric fields are primarily used in mathematical research, especially in areas such as algebra, number theory, and topology. They are less common in everyday language, making them chiefly relevant in written and academic contexts rather than oral speech.

Example Sentences

1. Ultrametric field provides a rigorous framework for discussions on p-adic numbers.
   The ultrametric field provides a rigorous framework for discussions on p-adic numbers.

2. The concept of an ultrametric field is essential for understanding certain algebraic structures.
   The concept of an ultrametric field is essential for understanding certain algebraic structures.

3. Researchers have shown that various properties can be deduced from the study of an ultrametric field.
   Researchers have shown that various properties can be deduced from the study of an ultrametric field.

Idiomatic Expressions

The term ultrametric field does not frequently appear in idiomatic expressions due to its technical nature. However, in academic discussions and mathematical contexts, related phrases may include:

1. Exploring the depths of an ultrametric field can lead to surprising mathematical truths.
   Exploring the depths of an ultrametric field can lead to surprising mathematical truths.

2. The study of an ultrametric field opens doors to new theories in number theory.
   The study of an ultrametric field opens doors to new theories in number theory.

3. In an ultrametric field, the relationships between elements can be quite unique.
   In an ultrametric field, the relationships between elements can be quite unique.

Etymology

The term ultrametric is derived from the prefix "ultra-", meaning "beyond" or "extreme," and "metric," relating to measurement. This reflects the nature of ultrametric spaces as extending typical metric spaces with additional properties.

Synonyms and Antonyms

Synonyms:
- Ultrametric space
- Non-Archimedean metric space

Antonyms:
- Metric space (in the context of standard distance functions)
- Archimedean space