archive-org.com » ORG » P » PLANETMATH.ORG

Total: 488

Choose link from "Titles, links and description words view":

Or switch to "Titles and links view".
  • canonical basis | planetmath.org
    of a number field adjusted canonical basis Related MinimalityOfIntegralBasis ExamplesOfRingOfIntegersOfANumberField ConditionForPowerBasis IntegralBasisOfQuadraticField CanonicalFormOfElementOfNumberField Major Section Reference Type of Math Object Theorem Parent integral basis Groups audience Buddy List of pahio Mathematics Subject Classification 11R04 no label found Add a correction Attach a problem Ask a question Comments Canonical basis Permalink Submitted by pahio on Fri 06 17 2005 22 47 If theta 3 2 what is like a canonical basis of Q theta Log in to post comments Re Canonical basis Permalink Submitted by Wkbj79 on Sun 10 08 2006 15 54 I noticed that no one has replied to this for over a year A canonical basis for Q theta where theta 3 2 is 1 theta theta 2 Using the notation of the entry to which the original post is attached all of the a ij s for i 1 are 0 and all of the d i s are 1 Log in to post comments Re Canonical basis Permalink Submitted by lalberti on Wed 03 26 2008 20 33 What is not said in the theorem the way it s written here is that the d i are unique That is if you find a basis that works then you ve found The basis well you still have some freedom as how to choose the a i j The real problem is not so much to find a canonical basis it s to find a basis Characterizing the integral closure of Z in a weird ring is the difficult part once you have a basis you ll easily find a way to torture it to make it canonical this is classical Z module algebra What you know for sure is that 1 theta tetha 2 are algebraic integers The question is did you forget some of them Q theta can be represented as a 3 dimensional vector space over Q and theta is an endomorphism with matrix 0 0 2 1 0 0 0 1 0 any element of Q theta is therefore a matrix a 2c 2b b a 2c a b theta c theta 2 c b a And its characteristic polynomial is I ll spare you the calculation x 3 3ax 2 3a 2 6bc x a 3 2b 3 4c 3 6abc The fact that the elements are in the integral closure of Z means that 3a 3a 2 6bc and a 3 2b 3 4c 3 6abc are all integers but a b and c are only rational numbers As I said before you may be missing generators of the integral closure but you have already 1 theta and theta 2 so you may restrict your search to new generators that lie in 0 1 3 that is you can suppose 0 a 1 0 b 1 and 0 c 1 This makes the determination of the factors possible though tedious as was computing the characteristic polynomial Ok let s see what happens in this case I ll just use my computer

    Original URL path: http://www.planetmath.org/canonicalbasis (2016-04-25)
    Open archived version from archive


  • canonical form of element of number field | planetmath.org
    irreducible the greatest common divisor of f x f x f x and b x b x b x is a constant polynomial which can be normed to 1 Thus there exist the polynomials φ x φ x varphi x and ψ x ψ x psi x of the ring ℚ x ℚ x mathbb Q x such that φ x f x ψ x b x 1 φ x f x ψ x b x 1 varphi x f x psi x b x equiv 1 Especially φ ϑ f ϑ 0 ψ ϑ b ϑ 1 φ ϑ subscript normal f ϑ absent 0 ψ ϑ b ϑ 1 varphi vartheta underbrace f vartheta 0 psi vartheta b vartheta 1 whence 1 b ϑ ψ ϑ 1 b ϑ ψ ϑ frac 1 b vartheta psi vartheta and consequently α a ϑ b ϑ a ϑ ψ ϑ ψ 1 ϑ α a ϑ b ϑ a ϑ ψ ϑ assign subscript ψ 1 ϑ alpha frac a vartheta b vartheta a vartheta psi vartheta psi 1 vartheta Hence α α alpha is a polynomial in ϑ ϑ vartheta with rational coefficients Let now ψ 1 x q x f x r x with deg r deg f n formulae sequence subscript ψ 1 x q x f x r x with deg r deg f n psi 1 x q x f x r x qquad textrm with mbox deg r mbox deg f n Denote r x c 0 c 1 x c n 1 x n 1 ℚ x assign r x subscript c 0 subscript c 1 x normal subscript c n 1 superscript x n 1 ℚ x r x c 0 c 1 x ldots c n 1 x n 1 in mathbb Q x It follows that α r ϑ c 0 c 1 ϑ c n 1 ϑ n 1 α r ϑ subscript c 0 subscript c 1 ϑ normal subscript c n 1 superscript ϑ n 1 alpha r vartheta c 0 c 1 vartheta ldots c n 1 vartheta n 1 whence 1 is true Suppose that we had also α s ϑ d 0 d 1 ϑ d n 1 ϑ n 1 α s ϑ subscript d 0 subscript d 1 ϑ normal subscript d n 1 superscript ϑ n 1 alpha s vartheta d 0 d 1 vartheta ldots d n 1 vartheta n 1 with every d i subscript d i d i rational This implies that c n 1 d n 1 ϑ n 1 c 1 d 1 ϑ c 0 d 0 0 subscript c n 1 subscript d n 1 superscript ϑ n 1 normal subscript c 1 subscript d 1 ϑ subscript c 0 subscript d 0 0 c n 1 d n 1 vartheta n 1 ldots c 1 d 1 vartheta c 0 d 0 0 i e that ϑ ϑ vartheta satisfies the equation c

    Original URL path: http://www.planetmath.org/canonicalformofelementofnumberfield (2016-04-25)
    Open archived version from archive

  • Cardano's formulae | planetmath.org
    2 1 i 3 2 frac 1 i sqrt 3 2 and u q 2 p 3 3 q 2 2 3 v q 2 p 3 3 q 2 2 3 formulae sequence assign u 3 q 2 superscript p 3 3 superscript q 2 2 assign v 3 q 2 superscript p 3 3 superscript q 2 2 displaystyle u sqrt 3 frac q 2 sqrt left frac p 3 right 3 left frac q 2 right 2 qquad v sqrt 3 frac q 2 sqrt left frac p 3 right 3 left frac q 2 right 2 3 The values of the cube roots must be chosen such that u v p 3 u v p 3 displaystyle uv frac p 3 4 Cardano s formulae essentially 2 and 3 were first published in 1545 in Geronimo Cardano s book Ars magna The idea of 2 and 3 is illustrated in the entry example of solving a cubic equation Let s now assume that the coefficients p p p and q q q are real The number of the real roots of 1 depends on the sign of the radicand R p 3 3 q 2 2 assign R superscript p 3 3 superscript q 2 2 displaystyle R left frac p 3 right 3 left frac q 2 right 2 of the above square root Instead of R R R we may use the discriminant D 108 R assign D 108 R D 108R of the equation As in examining the number of real roots of a quadratic equation we get three different cases also for the cubic 1 1 D 0 D 0 D 0 This is possible only when either p 0 p 0 p 0 or p q 0 p q 0 p q 0 Then we get the real roots y 1 2 q 2 3 subscript y 1 2 3 q 2 y 1 2 sqrt 3 q 2 y 2 y 3 q 2 3 subscript y 2 subscript y 3 3 q 2 y 2 y 3 sqrt 3 q 2 2 D 0 D 0 D 0 The square root R R sqrt R is real and one can choose for u u u and v v v the real values of the cube roots 3 these satisfy 4 Thus the root y 1 u v subscript y 1 u v y 1 u v is real and since y 2 3 u v 2 i 3 u v 2 subscript y 2 3 plus or minus u v 2 normal i 3 u v 2 y 2 3 frac u v 2 pm i sqrt 3 cdot frac u v 2 with u v u v u neq v the roots y 2 subscript y 2 y 2 and y 3 subscript y 3 y 3 are non real complex conjugates of each other 3 D 0 D 0 D 0 This requires that p p p

    Original URL path: http://www.planetmath.org/cardanosformulae (2016-04-25)
    Open archived version from archive

  • Cartesian coordinates | planetmath.org
    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 3A12D10 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 Cartesian coordinates The Cartesian coordinates of a point in ℝ 3 superscript ℝ 3 mathbb R 3 for determining its place in three dimensional space are the three real numbers x x x y y y and z z z which are called x x x coordinate or abscissa y y y coordinate or ordinate z z z coordinate or applicate The last name applicate is rare in English but its equivalents in continental European languages as die Applikate in German and aplikaat in Estonian are more known Similarly in ℝ n superscript ℝ n mathbb R n for all n 1 2 3 n 1 2 3 normal n 1 2 3 ldots one needs n n n coordinates for specifying the

    Original URL path: http://www.planetmath.org/cartesiancoordinates (2016-04-25)
    Open archived version from archive

  • Cassini oval | planetmath.org
    4 displaystyle x 2 y 2 c 2 2 4c 2 x 2 a 4 2 One sees that the curve is symmetric both in regard to x x x axis and in regard to y y y axis whence it suffices to examine it in the first quadrant x 0 x 0 x geqq 0 y 0 y 0 y geqq 0 If 2 is written as y 2 a 4 4 c 2 x 2 x 2 c 2 superscript y 2 superscript a 4 4 superscript c 2 superscript x 2 superscript x 2 superscript c 2 displaystyle y 2 sqrt a 4 4c 2 x 2 x 2 c 2 3 it appears that y y y is real only for a 4 4 c 2 x 2 x 2 c 2 superscript a 4 4 superscript c 2 superscript x 2 superscript x 2 superscript c 2 sqrt a 4 4c 2 x 2 geqq x 2 c 2 which condition can be simplified to x 2 c 2 a 2 superscript x 2 superscript c 2 superscript a 2 displaystyle x 2 c 2 leqq a 2 4 In order to y y y being real 4 gives the three cases 1 o superscript 1 normal o 1 underline o a c a c a c We have c 2 a 2 x c 2 a 2 superscript c 2 superscript a 2 x superscript c 2 superscript a 2 sqrt c 2 a 2 leqq x leqq sqrt c 2 a 2 thus the curve consists of two separate loops 2 o superscript 2 normal o 2 underline o a c a c a c Now 0 x c 2 0 x c 2 0 leqq x leqq c sqrt 2 the two loops meet in the origin the lemniscate of Bernoulli 3 o superscript 3 normal o 3 underline o a c a c a c Then 0 x c 2 a 2 0 x superscript c 2 superscript a 2 0 leqq x leqq sqrt c 2 a 2 there is one loop surrounding the origin psaxes Dx 1 Dy 1 0 0 2 7 1 5 2 7 1 6 psplot linecolor blue 1 4141 4144 x mul x mul 1 add sqrt x x mul sub 1 sub sqrt psplot linecolor blue 1 4141 4140 4 x mul x mul 1 add sqrt x x mul sub 1 sub sqrt sub psplot linecolor blue 1 48651 48654 x mul x mul 1 4641 add sqrt x x mul sub 1 sub sqrt psplot linecolor blue 1 48651 48650 4 x mul x mul 1 4641 add sqrt x x mul sub 1 sub sqrt sub psplot linecolor blue 1 7321 7324 x mul x mul 4 add sqrt x x mul sub 1 sub sqrt psplot linecolor blue 1 7321 7320 4 x mul x mul 4 add sqrt x x mul sub 1 sub sqrt sub psplot linecolor

    Original URL path: http://www.planetmath.org/cassinioval (2016-04-25)
    Open archived version from archive

  • casus irreducibilis | planetmath.org
    Primary tabs View active tab Coauthors PDF Source Edit casus irreducibilis Let the polynomial P x x n a 1 x n 1 a n assign P x superscript x n subscript a 1 superscript x n 1 normal subscript a n P x x n a 1 x n 1 ldots a n with complex coefficients a j subscript a j a j be irreducible i e irreducible in the field ℚ a 1 a n ℚ subscript a 1 normal subscript a n mathbb Q a 1 ldots a n of its coefficients If the equation P x 0 P x 0 P x 0 can be solved algebraically and if all of its roots are real then no root may be expressed with the numbers a j subscript a j a j using mere real radicals unless the degree n n n of the equation is an integer power of 2 References 1 K Väisälä Lukuteorian ja korkeamman algebran alkeet Tiedekirjasto No 17 Kustannusosakeyhtiö Otava Helsinki 1950 Keywords irreducible polynomial roots real Related RadicalExtension CardanosFormulae TakingSquareRootAlgebraically EulersDerivationOfTheQuarticFormula Type of Math Object Theorem Major Section Reference Parent radical extension Groups audience Buddy List of pahio Mathematics Subject Classification 12F10 no label found Add a correction Attach a problem Ask a question Comments Casus irreducibilis Permalink Submitted by rm50 on Fri 12 28 2007 22 42 Jussi I m not sure I understand or believe the current statement Do you mean that if K is a finite extension of Q and f x in K x can be solved algebraically and has real roots then no radical extension of K contains all the roots of f x unless f has degree a power of 2 Roger Log in to post comments Re Casus irreducibilis Permalink Submitted by mathcam on Sat 12 29 2007 17 09 In fact K need not be finite over Q if say one of the coefficients is pi I agree that something is fishy here Cam Log in to post comments Re Casus irreducibilis Permalink Submitted by mathcam on Sat 12 29 2007 17 22 Jussi I m not sure I understand or believe the current statement Do you mean that if K is a finite extension of Q and f x in K x can be solved algebraically and has real roots then no radical extension of K contains all the roots of f x unless f has degree a power of 2 Ah the statement is slightly different Let s suppose that f has rational coefficients for simplicity Then I think it should be If f in Q x is irreducible has all real roots and has degree not a power of 2 then the real roots of f are not contained in any real radical extension of Q This sounds more plausible maybe real should be replaced with totally real not sure and brings us back to the motivating phenomenon of casus irreducibilis for cubics these roots are real numbers which necessitate

    Original URL path: http://www.planetmath.org/casusirreducibilis (2016-04-25)
    Open archived version from archive

  • catacaustic | planetmath.org
    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 3A12F10 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 catacaustic Given a plane curve γ γ gamma its catacaustic Greek ϰ α τ α ϰ α υ σ τ ι ϰ o ς ϰ α τ normal α ϰ α υ σ τ ι ϰ normal o ς varkappa alpha tau acute alpha varkappa alpha upsilon sigma tau iota varkappa acute o varsigma burning along is the envelope of a family of rays reflected from γ γ gamma after having emanated from a fixed point which may be infinitely far in which case the rays are initially parallel For example the catacaustic of a logarithmic spiral reflecting the rays emanating from the origin is a congruent spiral The catacaustic of the exponential curve y e x y superscript e x y e x reflecting the vertical rays x t x t x t is the catenary y cosh x 1

    Original URL path: http://www.planetmath.org/catacaustic (2016-04-25)
    Open archived version from archive

  • Catalan's conjecture | planetmath.org
    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 3A51N20 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 3A26B05 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 3A26B05 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 3A26A24 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

    Original URL path: http://www.planetmath.org/catalansconjecture (2016-04-25)
    Open archived version from archive



  •