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  • Now suppose that <math>n > 0</math>. Let <math>C</math> be a [[splitting field]] of <math>F</math> over <math>\mathbb{C}</math>, and let <math>x_1, \dotsc ...e both complex numbers, so <math>x_i</math> and <math>x_j</math> satisfy a quadratic equation with complex coefficients. Hence they are complex numbers. There
    5 KB (832 words) - 13:22, 11 January 2024
  • ...e is the [[real number | real]], the [[complex number]]s or any abstract [[field]] is a value <math>a</math> in the [[domain]] of the function such that <ma ...s <cmath>(x - (a + bi))(x - (a - bi)) = x^2 - 2ax + (a^2 + b^2),</cmath> a quadratic polynomial with real coefficients. As such, dividing through by the product
    8 KB (1,427 words) - 20:37, 13 March 2022
  • <cmath> \genfrac{(}{)}{}{}{a}{p} =\begin{cases} 1 & \text{if } a \text{ is a quadratic residue modulo } p, \ We say that <math>a</math> is a '''quadratic residue''' modulo <math>p</math> if there exists an integer <math>n</math>
    7 KB (1,182 words) - 15:46, 28 April 2016
  • ...'' relates the rank of the [[abelian group]] of [[point]]s over a [[number field]] of an [[elliptic curve]] <math>E</math> to the [[order]] of the [[root|ze certain [[quadratic field]]s (Henri Darmon, of McGill University). It has been an open problem for ar
    7 KB (1,102 words) - 16:23, 6 September 2008
  • A rectangular field is half as wide as it is long and is completely enclosed by <math>x</math> If in applying the quadratic formula to a quadratic equation
    23 KB (3,646 words) - 20:53, 21 June 2024
  • ...millennium in May 2000. The problems all have significant impacts on their field of mathematics and beyond, and were all unsolved at the time of the offerin ...re relates the rank of the [[abelian group]] of [[point]]s over a [[number field]] of an [[elliptic curve]] <math>E</math> to the [[order]] of the [[root|ze
    13 KB (1,969 words) - 16:57, 22 February 2024
  • ...es, Tony gets bored and walks to the creek a few yards behind the baseball field. One of Tony's classmates One day when Wendy is riding her horse Vanessa, they get to a field where some tourists are following Martin (the tour guide) on some horses. M
    71 KB (11,749 words) - 11:39, 20 November 2024
  • The theorem is a starting point in the investigation of [[binary quadratic forms]]. Historically, two of the main questions have been: Since 0 and 1 are the only [[quadratic residue]]s mod 4, it follows that if <math>p</math> is a prime number repre
    4 KB (612 words) - 20:31, 7 January 2025
  • This condition can be rephrased in terms of [[field theory]] as follows: ...lpha)</math> such that each extension <math>K_i\subseteq K_{i+1}</math> is quadratic (i.e. <math>[K_{i+1}:K_i]=2</math>).
    8 KB (1,305 words) - 07:39, 21 August 2009
  • ...^2,\ldots,\zeta_n^{n-1}</math>. As the set of constructible numbers is a [[field]], these numbers will be constructible iff <math>\zeta_n</math> is construc ...ta_n)</math> such that each extension <math>K_i\subseteq K_{i+1}</math> is quadratic (i.e. <math>[K_{i+1}:K_i]=2</math>). We claim that this happens iff <math>\
    5 KB (926 words) - 17:47, 4 March 2022
  • In solving a problem that reduces to a quadratic equation one student makes a mistake only in the constant term of the equat ...es of a rectangular field, a boy took a shortcut along the diagonal of the field and
    21 KB (3,123 words) - 13:24, 20 February 2020
  • ...ber|algebraic numbers]] and structures involving them, especially [[number field|algebraic number fields]]. ...e responsible for the successful attack on [[Fermat's Last Theorem]]. This field is extremely rich and advanced (indeed, prerequisites to courses in this to
    10 KB (1,646 words) - 14:04, 28 May 2020
  • ...adjoining a root of <math>x^2-2=0</math> to <math>\mathbb F_5</math>, the field with <math>5</math> elements. ...\beta^2</math> in <math>\mathbb F_5</math> (i.e. <math>3</math> would be a quadratic residue modulo <math>5</math>, and it's not).
    1 KB (257 words) - 14:58, 29 January 2021
  • We can use the process of elimination to narrow down the field substantially: ...\text{always increases as } x\text{ increases}</math> is wrong due to the quadratic nature of the function.
    1 KB (216 words) - 16:25, 1 August 2020
  • ...s can the farmer choose crops to plant in each of the four sections of the field? ...h> is satisfied by exactly three real numbers. Among all the disrespectful quadratic polynomials, there is a unique such polynomial <math>\tilde{p}(x)</math> fo
    14 KB (2,164 words) - 23:36, 19 December 2024
  • Let <math>p</math> be an odd prime. An integer <math>x</math> is called a <i>quadratic non-residue</i> if <math>p</math> does not divide <math>x - t^2</math> for ...ath>1 \le a < p</math>, and both <math>a</math> and <math>4 - a</math> are quadratic non-residues. Calculate the remainder when the product of the elements of <
    4 KB (828 words) - 13:27, 4 December 2024