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)) are categorized as such.
- This implies that any two uncountable algebraically closed fields of the same cardinality and the same characteristic are isomorphic.
Baseball players field a ball, and you need nine players to field a team. All the subjects you study in school are different fields of study. This word has many meanings — such as a field of daffodils (a field of study), or a field of battle in a war.
Definition
Substituting x for X in rational fractions yields this isomorphism. The extension degree E(x) / E (representing E’s dimension as an E-vector space), corresponds 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 when and only when 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 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, https://ambassadorsevents.com 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.
The Lefschetz principle states that C is elementarily equivalent to any algebraically closed field F of characteristic zero. In model theory (a branch of mathematical logic), two fields E and F are called elementarily equivalent if every mathematical statement that is true for E is also true for F and conversely. This implies that any two uncountable algebraically closed fields of the same cardinality and the same characteristic are isomorphic. The latter is defined as the maximal number of elements in F that are algebraically independent over the prime field. The latter condition is always satisfied if E has characteristic 0. For such an extension, being normal and separable means that all zeros of f are contained in F and that f has only simple zeros.
After cleanly fielding the ball, Tucker moved back toward third base. On the island of Oahu — Hawaii, field trials were carried out on a residential street. To demonstrate how words are used in real contexts — examples are supplied. Begin your educational journey now with our engaging library of themed word lists crafted by the specialists at Vocabulary.com – we’ll assist you in maximizing your study time! Explore this curated, interactive word list created by our team of English language experts at Vocabulary.com – part of a collection exceeding 17,000 lists designed for learners globally!
A cultivated expanse of land, especially one devoted to a particular crop Field refers to an open area of land, usually used for agriculture or sports. The correct spelling is “Field,” while the incorrect spelling is “Feild.” A field is an open area of land or a specialized domain of knowledge or activity. Definitions and idiom definitions from Dictionary.com Unabridged (based on the Random House Unabridged Dictionary), © Random House, Inc. 2023
The algebraic closure Q of Q is referred to as the field encompassing algebraic numbers, for example. A field is termed an algebraic closure of F if it is algebraic over F (generally speaking — not excessively large relative to F) and is algebraically closed (sufficiently large to provide solutions to all polynomial equations). Since the equation doesn’t hold, neither the rational nor the real numbers are algebraically closed.
Engaging in work or study under real-world conditions, away from a laboratory or office setting. The away team introduced two new players along with the backup goalkeeper. Given the depth of talent, France could have possibly fielded a B team in this World Cup and still advanced to the quarterfinals.
Alternatively — a field can be defined using four binary operations (addition, subtraction, multiplication, and division) along with their essential properties. The following properties, known as field axioms, must be satisfied by these operations. The addition of a and b produces a result called the sum of a and b, represented as a + b. Formally (a field consists of a set F paired with two binary operations on F), known as addition and multiplication, which adhere to the axioms outlined below.
