Difference between revisions of "Space"
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In mathematics, a [[w:field (mathematics)|field]] <math>F</math> is said to be [[w:algebraically closed|algebraically closed]] if every polynomial in one variable of degree at least <math>1</math>, with [[w:coefficient|coefficient]]s in <math>F</math>, has a zero ([[w:root (mathematics)|root]]) in <math>F</math>. | In mathematics, a [[w:field (mathematics)|field]] <math>F</math> is said to be [[w:algebraically closed|algebraically closed]] if every polynomial in one variable of degree at least <math>1</math>, with [[w:coefficient|coefficient]]s in <math>F</math>, has a zero ([[w:root (mathematics)|root]]) in <math>F</math>. | ||
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| + | ;What is a ''field''? | ||
| + | In [[w:abstract algebra|abstract algebra]], a [[w:field|field]] is an [[w:algebraic structure|algebraic structure]] in which the operations of addition, subtraction, multiplication and [[w:division (mathematics)|division]] (except division by zero) may be performed, and the same rules hold which are familiar from the [[w:arithmetic|arithmetic]] of ordinary [[w:number|number]]s. | ||
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| + | The mathematical discipline concerned with the study of fields is called [[w:field theory (mathematics)|field theory]]. | ||
Revision as of 03:08, 26 October 2006
- Organising some notes about spacetime fundamentals from pedia
- The fundamental theorem of algebra
In mathematics, the fundamental theorem of algebra states that every complex polynomial [math]p(z)[/math] in one variable and of degree [math]n[/math] ≥ [math]1[/math] has some complex root. In other words, the field of complex numbers is algebraically closed and therefore, as for any other algebraically closed field, the equation [math]p(z)=0[/math] has [math]n[/math] roots (not necessarily distinct).
The name of the theorem is now considered a misnomer by many mathematicians, since it is an instance of analysis rather than algebra.
- Algebraically closed field
In mathematics, a field [math]F[/math] is said to be algebraically closed if every polynomial in one variable of degree at least [math]1[/math], with coefficients in [math]F[/math], has a zero (root) in [math]F[/math].
- What is a field?
In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers.
The mathematical discipline concerned with the study of fields is called field theory.
