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  • characteristic of finite ring | planetmath.org
    3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A00A05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit characteristic of finite ring The characteristic of the residue class ring ℤ m ℤ ℤ m ℤ mathbb Z m mathbb Z which contains m m m elements is m m m too More generally one has the Theorem The characteristic of a finite ring divides the number of the elements of the ring Proof Let n n n be the characteristic of the ring R R R with m m m elements Since m m m is the order of the group R R R the Lagrange s theorem implies that m a 0 a R formulae sequence m a 0 for all a R ma 0 quad forall a in R Let m q n r m q n r m qn r where 0 r n 0 r n 0 leqq r n Because r a m q n a m a q n a 0 0 0 a R formulae sequence r a m q n a m a q n a 0 0 0 for all a R ra m qn a ma q na 0 0 0 quad forall a in R and n

    Original URL path: http://www.planetmath.org/characteristicoffinitering (2016-04-25)
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  • characteristic polynomial of algebraic number | planetmath.org
    x its minimal polynomial and ϑ 1 ϑ ϑ 2 ϑ n subscript ϑ 1 ϑ subscript ϑ 2 normal subscript ϑ n vartheta 1 vartheta vartheta 2 ldots vartheta n its algebraic conjugates Let α α alpha be an element of the number field ℚ ϑ ℚ ϑ mathbb Q vartheta and r x c 0 c 1 x c n 1 x n 1 assign r x subscript c 0 subscript c 1 x normal subscript c n 1 superscript x n 1 r x c 0 c 1 x ldots c n 1 x n 1 the canonical polynomial of α α alpha with respect to ϑ ϑ vartheta We consider the numbers r ϑ 1 α α 1 r ϑ 2 α 2 r ϑ n α n formulae sequence r subscript ϑ 1 α assign superscript α 1 formulae sequence assign r subscript ϑ 2 superscript α 2 normal assign r subscript ϑ n superscript α n displaystyle r vartheta 1 alpha alpha 1 quad r vartheta 2 alpha 2 quad ldots quad r vartheta n alpha n 1 and form the equation g x i 1 n x r ϑ i x α 1 x α 2 x α n x n g 1 x n 1 g n 0 assign g x superscript subscript product i 1 n x r subscript ϑ i x superscript α 1 x superscript α 2 normal x superscript α n superscript x n subscript g 1 superscript x n 1 normal subscript g n 0 g x prod i 1 n x r vartheta i x alpha 1 x alpha 2 cdots x alpha n x n g 1 x n 1 ldots g n 0 the roots of which are the numbers 1 and only these The coefficients g i subscript g i g i of the polynomial g x g x g x are symmetric polynomials in the numbers ϑ 1 ϑ 2 ϑ n subscript ϑ 1 subscript ϑ 2 normal subscript ϑ n vartheta 1 vartheta 2 ldots vartheta n and also symmetric polynomials in the numbers α i superscript α i alpha i The fundamental theorem of symmetric polynomials implies now that the symmetric polynomials g i subscript g i g i in the roots ϑ i subscript ϑ i vartheta i of the equation f x 0 f x 0 f x 0 belong to the ring determined by the coefficients of the equation and of the canonical polynomial r x r x r x thus the numbers g i subscript g i g i are rational whence the degree of α α alpha is at most equal to n n n It is not hard to show see the entry degree of algebraic number of that the degree k k k of α α alpha divides n n n and that the numbers 1 consist of α α alpha and its algebraic conjugates α 2 α k subscript α 2

    Original URL path: http://www.planetmath.org/characteristicpolynomialofalgebraicnumber (2016-04-25)
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  • characterizations of integral | planetmath.org
    jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A12F05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11R04 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A11R04 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit characterizations of integral Theorem Let R R R be a subring of a field K K K 1 R 1 R 1 in R and let α α alpha be a non zero element of K K K The following conditions are equivalent 1 α α alpha is integral over R R R 2 α α alpha belongs to R α 1 R superscript α 1 R alpha 1 3 α α alpha is unit of R α 1 R superscript

    Original URL path: http://www.planetmath.org/characterizationsofintegral (2016-04-25)
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  • Chinese remainder theorem in terms of divisor theory | planetmath.org
    2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A13B21 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A13B21 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit Chinese remainder theorem in terms of divisor theory In a ring with a divisor theory a congruence α β mod α annotated β pmod alpha equiv beta mathop rm mod mathfrak a with respect to a divisor module mathfrak a means that α β fragments a normal α β mathfrak a mid alpha beta Theorem Let mathcal O be an integral domain having the divisor theory normal superscript mathcal O to mathfrak D For arbitrary pairwise coprime divisors 1 s subscript 1 normal subscript s mathfrak a 1 ldots mathfrak a s in mathfrak D and for arbitrary elements α 1 α s subscript α 1 normal subscript α s alpha 1 ldots alpha s of the domain mathcal O there exists an element ξ ξ xi in mathcal O such that ξ α 1 mod 1 ξ α s mod s ξα1a1mod ξαsasmod displaystyle begin cases xi equiv alpha 1 mathop rm mod mathfrak a 1 cdots qquad cdots qquad cdots xi equiv alpha s mathop rm mod mathfrak a s end cases Proof Let i j i j i 1 s fragments subscript i assign subscript product j i subscript j fragments normal i 1 normal normal normal s normal normal mathfrak b i prod j neq i mathfrak a j quad i 1 ldots s Apparently the divisors 1 s subscript 1 normal subscript s mathfrak b 1 ldots mathfrak b s are mutually coprime whence there are in the ring mathcal O the elements β 1 β s subscript β 1 normal subscript β s beta 1

    Original URL path: http://www.planetmath.org/chineseremaindertheoremintermsofdivisortheory (2016-04-25)
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  • circular segment | planetmath.org
    sparql sparql module User error Socket error Could not connect to http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A13A05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D proxy 0 Connection refused in ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in getFormat via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A13A05 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit circular segment A chord of a circle divides the corresponding disk into two circular segments The perimetre of a circular segment consists thus of the chord c c c and a circular arc a a a The magnitude r r r of the radius of circle and the magnitude α α alpha of a central angle naturally determine uniquely the magnitudes of the corresponding arc and chord and these may be directly calculated from a r α c 2 r sin α 2 a rα c 2rsinα2 displaystyle begin cases a r alpha c 2r sin frac alpha 2 end cases 1 Conversely the magnitudes of a a a and c c c a absent a a uniquely determine r r r and α α alpha from the pair of equations 1 but r r r and α α alpha are generally not expressible in a closed form this becomes clear from the relationship c a α 2 sin α 2 normal c a α 2 α 2 frac c a cdot frac alpha 2 sin frac alpha 2 implied by 1 α α alpha a a a c c c r r r The area of a circular segment is obtained by subtracting from resp adding to the area of the corresponding sector the area of the isosceles triangle having the chord as base the

    Original URL path: http://www.planetmath.org/circularsegment (2016-04-25)
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  • cissoid of Diocles | planetmath.org
    PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51M04 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit cissoid of Diocles Let c c c be a circle with diameter O A a O A a OA a Set a tangent line t t t of the circle at the point A A A For any point C C C of c c c let P P P be the intersection point of the secant line O C O C OC and the tangent line t t t Determine on the secant line between O O O and P P P the point Q Q Q such that P Q O C P Q O C PQ OC Then the locus of the point Q Q Q is the cissoid of Diocles The name is derived from Greek ϰ ι σ σ o ς ϰ ι σ σ normal o ς varkappa iota sigma sigma acute o varsigma kissos ivy ε ι δ o ς ε ι δ o ς varepsilon iota delta o varsigma eidos form kind type The cissoid is symmetric with regard to the line O A O A OA having at O O O a cusp The line t t t is the asymptote of the curve For deriving the equation of the cissoid chose the ray O A O A OA for the positive x x x axis Let φ φ varphi be the slope angle polar angle of any C C C on c c c From the triangle O A P O A P OAP we see that O P a cos φ O P a φ displaystyle OP frac a cos varphi Since A C O normal A C O angle ACO is a right angle we have O C a cos φ O C a φ OC a cos varphi It follows that O Q r a cos φ a cos φ O Q r a φ a φ displaystyle OQ r frac a cos varphi a cos varphi that is r a sin φ tan φ r a φ φ displaystyle r a sin varphi tan varphi 1 For obtaining the equation in rectangular coordinates we may write 1 as r 2 r sin φ a tan φ superscript r 2 r φ a φ r 2 r sin varphi a tan varphi i e x 2 y 2 y a y x superscript x 2 superscript y 2 normal y a y x displaystyle x 2 y

    Original URL path: http://www.planetmath.org/cissoidofdiocles (2016-04-25)
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  • Clairaut's equation | planetmath.org
    beta sites all modules sparql sparql module User error missing stream in readStream http planetmath org 8890 sparql query 0APREFIX msc 3A 3Chttp 3A 2F 2Fmsc2010 org 2Fresources 2FMSC 2F2010 2F 3E PREFIX skos 3A 3Chttp 3A 2F 2Fwww w3 org 2F2004 2F02 2Fskos 2Fcore 23 3E PREFIX dct 3A 3Chttp 3A 2F 2Fpurl org 2Fdc 2Fterms 2F 3E PREFIX local 3A 3Chttp 3A 2F 2Flocal virt 2F 3E SELECT 3Flabel WHERE 7B GRAPH 3Chttp 3A 2F 2Flocalhost 3A8890 2FDAV 2Fhome 2Fpm 2Frdf sink 23this 3E 7B msc 3A51 00 skos 3AprefLabel 3Flabel FILTER langMatches 28 lang 28 3Flabel 29 2C 22en 22 29 7D 7D via ARC2 Reader in sparql request line 92 of home jcorneli beta sites all modules sparql sparql module Primary tabs View active tab Coauthors PDF Source Edit The ordinary differential equation y x d y d x ψ d y d x y x d y d x ψ d y d x displaystyle y x frac dy dx psi left frac dy dx right 1 where ψ ψ psi is a given differentiable real function is called Clairaut s equation For solving the equation we use an auxiliary variable p d y d x fragments p normal d y d x p frac dy dx and write 1 as y p x ψ p y p x ψ p y px psi p Differentiating this equation gives p x d p d x p ψ p d p d x p x d p d x p superscript ψ normal p d p d x p x frac dp dx p psi prime p frac dp dx or x ψ p d p d x 0 x superscript ψ normal p d p d x 0 x psi prime p frac dp dx 0 The zero rule of product now yields the alternatives d p d x 0 d p d x 0 displaystyle frac dp dx 0 2 and x ψ p 0 x superscript ψ normal p 0 displaystyle x psi prime p 0 3 Integrating 2 we get p C p C p C constant and substituting this in 1 gives the general solution y C x ψ C y C x ψ C displaystyle y Cx psi C 4 which presents a family of straight lines If 3 allows to solve p p p in terms of x x x p p x p p x p p x we can write 1 as y x p x ψ p x y x p x ψ p x displaystyle y xp x psi p x 5 which is easy to see satisfying 1 The solution 5 may not be gotten from 4 using any value of C C C It is a singular solution which may be obtained by eliminating the parameter p p p from the equations y p x ψ p x ψ p 0 formulae sequence y p x ψ p x superscript ψ normal p

    Original URL path: http://www.planetmath.org/clairautsequation (2016-04-25)
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  • class numbers of imaginary quadratic fields | planetmath.org
    1 1 1 1 1 47 47 47 5 5 5 97 97 97 4 4 4 146 146 146 16 16 16 2 2 2 1 1 1 51 51 51 2 2 2 101 101 101 14 14 14 149 149 149 14 14 14 3 3 3 1 1 1 53 53 53 6 6 6 102 102 102 4 4 4 151 151 151 7 7 7 5 5 5 2 2 2 55 55 55 4 4 4 103 103 103 5 5 5 154 154 154 8 8 8 6 6 6 2 2 2 57 57 57 4 4 4 105 105 105 8 8 8 155 155 155 4 4 4 7 7 7 1 1 1 58 58 58 2 2 2 106 106 106 6 6 6 157 157 157 6 6 6 10 10 10 2 2 2 59 59 59 3 3 3 107 107 107 3 3 3 158 158 158 8 8 8 11 11 11 1 1 1 61 61 61 6 6 6 109 109 109 6 6 6 159 159 159 10 10 10 13 13 13 2 2 2 62 62 62 8 8 8 110 110 110 12 12 12 161 161 161 16 16 16 14 14 14 4 4 4 65 65 65 8 8 8 111 111 111 8 8 8 163 163 163 1 1 1 15 15 15 2 2 2 66 66 66 8 8 8 113 113 113 8 8 8 165 165 165 8 8 8 17 17 17 4 4 4 67 67 67 1 1 1 114 114 114 8 8 8 166 166 166 10 10 10 19 19 19 1 1 1 69 69 69 8 8 8 115 115 115 2 2 2 167 167 167 11 11 11 21 21 21 4 4 4 70 70 70 4 4 4 118 118 118 6 6 6 170 170 170 12 12 12 22 22 22 2 2 2 71 71 71 7 7 7 119 119 119 10 10 10 173 173 173 14 14 14 23 23 23 3 3 3 73 73 73 4 4 4 122 122 122 10 10 10 174 174 174 12 12 12 26 26 26 6 6 6 74 74 74 10 10 10 123 123 123 2 2 2 177 177 177 4 4 4 29 29 29 6 6 6 77 77 77 8 8 8 127 127 127 5 5 5 178 178 178 8 8 8 30 30 30 4 4 4 78 78 78 4 4 4 129 129 129 12 12 12 179 179 179 5 5 5 31 31 31 3 3 3 79 79 79 5 5 5 130 130 130 4 4 4 181 181 181 10 10 10 33 33 33 4 4 4 82 82 82 4 4 4 131 131 131 5 5 5 182 182 182

    Original URL path: http://www.planetmath.org/classnumbersofimaginaryquadraticfields (2016-04-25)
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