A geographic region (land or sea) under which something valuable is found; A piece of land of considerable size; esp. — a piece inclosed for tillage or pasture. Cleared land; land suitable for tillage or pasture; cultivated ground; the open country. To be the team catching and throwing the ball — as opposed to hitting it. A land area free of woodland, cities, and towns; an area of open country. A portion of land or a geologic formation containing a specified natural resource
Lists of vocabulary that include the term field
The Artin–Schreier theorem states that a field can be ordered if and only if it is a formally real field (which means that any quadratic equation Since fields are ubiquitous in mathematics and beyond), several refinements of the concept have been adapted to the needs of particular mathematical areas. For any algebraically closed field F of characteristic 0, the algebraic closure of the field F((t)) of Laurent series is the field of Puiseux series, obtained by adjoining roots of t. It is commonly referred to as the algebraic closure and denoted F. Any field F has an algebraic closure, which is moreover unique up to , non-unique, isomorphism.
Examples of field in a Sentence
- In mathematics (a field is a set on which addition), subtraction, multiplication, and division are defined and behave as the corresponding operations on rational numbers do.
- The best known fields are the field of rational numbers (the field of real numbers), and the field of complex numbers.
- A pivotal notion in the study of field extensions F / E are algebraic elements.
- By definition, number fields , finite extensions of Q, or function fields over Fq (finite extensions of Fq(t)) exist in this category.
- This implies that any two uncountable algebraically closed fields of the same cardinality and the same characteristic are isomorphic.
To form a team (nine players are needed), and they field a baseball. The various subjects you encounter in school represent distinct areas of study. This term can signify many concepts (including a field of daffodils), a field of inquiry, or a battlefield in conflict.
Definition
By replacing x with X in rational fractions, one obtains this isomorphism. Furthermore — the extension E(x) / E’s degree, also known as the dimension of E(x) as an E-vector space, is equal to the smallest degree n of a polynomial equation involving x, as previously mentioned. An algebraic extension of E is formed by the subfield E(x) (generated by an element x), if and only if x is algebraic.
Informally — a field is a set with an addition operation a + b and a multiplication operation a ⋅ b that behave as they do for rational numbers and real https://ambassadorsevents.com numbers. Function fields can help describe properties of geometric objects. This includes different branches of mathematical analysis, which are based on fields with additional structure. Fields serve as foundational notions in several mathematical domains. Galois theory, devoted to understanding the symmetries of field extensions, provides an elegant proof of the Abel–Ruffini theorem that general quintic equations cannot be solved in radicals.
This function field analogy can help to shape mathematical expectations (often first by understanding questions about function fields), and later treating the number field case. They are — by definition, number fields (finite extensions of Q) or function fields over Fq (finite extensions of Fq(t)). The study of function fields and their geometric meaning in higher dimensions is referred to as birational geometry. The function field is invariant under isomorphism and birational equivalence of varieties. In this case, one considers the algebra of holomorphic functions, i.e., complex-valued differentiable functions.
It is thus customary to speak of the finite field with q elements — denoted by Fq or GF(q). By contrast, in F2, f has only two zeros , namely 0 and 1,, so f does not split into linear factors in this smaller field. Such a splitting field is an extension of Fp in which the polynomial f has q zeros.
According to the Lefschetz principle, C is elementarily equivalent to any algebraically closed field F that has a characteristic of zero. In model theory (which is a segment of mathematical logic), two fields E and F are termed elementarily equivalent if every true mathematical statement for E also holds for F, and vice versa. This means that any two uncountable algebraically closed fields sharing the same cardinality and characteristic are isomorphic. The latter refers to the largest number of elements in F that are algebraically independent over the prime field. If E possesses a characteristic of 0, the latter condition is invariably fulfilled. For such an extension, being separable and normal entails that all roots of f reside in F and that f consists solely of simple roots.
When he fielded it cleanly, Tucker shuffled back toward third base. Field trials were conducted on a residential road on the island of Oahu, Hawaii. Examples are provided to illustrate real-world usage of words in context. Start your learning journey today with our library of interactive, themed word lists built by the experts at Vocabulary.com – we’ll help you make the most of your study time! Check out this interactive, curated word list from our team of English language specialists at Vocabulary.com – one of over 17,000 lists we’ve built to help learners worldwide!
A cultivated area of land (particularly one designated for a specific crop), is referred to as a field, which signifies an open land space typically utilized for agricultural or sporting purposes. Field” is the correct spelling, while “Feild” is a misspelling. A field can be defined as either an open area of land or a specialized area of expertise or activity. Definitions and idiomatic meanings are provided by Dictionary.com Unabridged (drawing from the Random House Unabridged Dictionary), © Random House, Inc. 2023.
For example — the algebraic closure Q of Q is called the field of algebraic numbers. A field containing F is called an algebraic closure of F if it is algebraic over F , roughly speaking, not too big compared to F, and is algebraically closed (big enough to contain solutions of all polynomial equations). The rational and the real numbers are not algebraically closed since the equation
Working or studying in real-world conditions, outside of a laboratory or office. The away team fielded two new players and the second-choice goalkeeper. The talent pool there is so deep, France probably could have fielded a B team in this World Cup and made it to the quarterfinals.
One can alternatively define a field by four binary operations (addition (subtraction), multiplication, and division) and their required properties. These operations are required to satisfy the following properties, called field axioms. The result of the addition of a and b is called the sum of a and b, and is denoted a + b. Formally (a field is a set F together with two binary operations on F), called addition and multiplication, satisfying the axioms given below.
