1. Pure and Applied Number Theory School 2015.8.11 Cheolmin Park NIMS

2.  SAGE (System for Algebra and Geometry Experimentation): open-source mathematics software

3.  Sage is a free open-source mathematics software system licensed under the GPL. It combines the power of many existing open- source packages into a common Python- based interface. ◦ Mission: Creating a viable free open source alternative to Magma, Maple, Mathematica and Matlab.  http://www.sagemath.org/ http://www.sagemath.org/  Current version: 6.8  Free, open source

4.  Sage provides ◦ Basic Algebra and Calculus  Solving Equations, Differentiation, … ◦ Plotting 2,3-dimensional plots ◦ Linear Algebra, Polynomials, Groups ◦ p-adic numbers ◦ Elliptic Curve, …  Easy to use

5.  SAGE is Linux based software.  If you want to use SAGE on Window OS, you need to install VirtualBox for Window which can run linux in Window OS.  Refer to “install guide” in www.sagemath.org/ www.sagemath.org/  We use SAGE Online!!! ◦ https://cloud.sagemath.com/ https://cloud.sagemath.com/

7.  IE9 cannot connect to this site.  In this case, use higher version of IE or chrome browser.  If you already have account, sign in  If not, you need to create account. ◦ It is for free. ◦ Click item “First, agree to the Terms of Service”Terms of Service ◦ Write Name/Email/ Choose a password ◦ Click item “Create account for free”

8.  Create New Project…  Title…and click “create project”  Click item “ ⨁New”

9.  Click “Sage Worksheet”

10.  A very brief Tour of Sage

11.  Run code: click or shift+Enter ◦ e.g. 2+3 shift+Enter  Sage uses = for assignment  Comparison check: ==, =,  Basic arithmetic ◦ **, ^: exponentiation ( 2^3 == 2**3) ◦ %: remainder (10%3) ◦ //: quotient ◦ sqrt(10), sin(5), sin(pi)

12.  Sage use for calling function ◦ variable.function() or function(variable) ◦ (eg1) n=2015 n.factor() factor(n) ◦ (eg2) M=matrix(2, 2, [1,pi, e,5]) M.det() det(M)  For defining function, (eg1)def f(a,b): return a+b (eg2)def fm(a,b): if a >=b: c=a else: c=b return c Note: indentation is important.

13.  var('x y')  f = x^3 * e^(y*x) * sin(y*x);  f.diff(x)  latex(f.diff(x))  show(f.diff(x))  f(x,3): You can evaluate f at y=3  plot(f(x,3))

14.  CC: complex field  RR: real field  QQ: rational field  ZZ: integer ring  Integers(n): ring Z/nZ of integers modulo n  GF(p): finite field of order p  GF(p^n): finite field of order p^n  QQ[x]: Univariate Polynomial Ring in x over Rational Field  QQ[x,y]: Multivariate Polynomial Ring in x, y over Rational Field

15.  k. = CC[] (k is univariate polynomial ring in x over complex field)  f=x^3+x  factor(f)  f.roots()  Change field and try factor(f), f.roots()

16.  EllipticCurve(fields(rings),[a1,a2,a3,a4,a5]): Elliptic Curve defined by y^2 + a1*x*y + a3*y = x^3 + a2*x^2 + a4*x + a5 over Fields  Example ◦ EllipticCurve(QQ,[10,20,30,40,50]) ◦ k=EllipticCurve(GF(127),[10,20,30,40,50]) ◦ k.order(), P=k.random_point(), Q=k(75,4) ◦ 123*P, P+Q

17.  mod(8+5, 7) == (8+5)%7 : (8+5) mod 7  pow(3,6,11) : 3^6 mod 11  inverse_mod(2,19): 2^-1 mod 19  gcd(12, 36); xgcd(3,5)  functions related to prime ◦ prime_factor(n), divisors(n), next_prime(n), nth_prime(i), is_prime(n), prime_pi(n) (# of primes less than n), euler_phi(n), primitive_root(p)…  x=crt(y1,y2,p1,p2): x mod p1 =y1, x mod p2 =y2

18.  P. =QQ[]  K. =NumberField(x^2+1)  R. = K[]  f = y^2 + y + 1  L. = K.extension(f)  S. =CyclotomicField(n)

19.  P. =QQ[]  K. =NumberField(x^2+1)  OK=K.ring_of_integers()  O3 = K.order(3*a); O3 #Z+3aZ  O3.gens()

20.  K. = NumberField(x^2 + 23)  I = K.fractional_ideal(2, 1/2*a - 1/2)  J = I^2  I*J  factor(I) factor(J)  is_prime(I)  I = K.fractional_ideal(2)  I.factor()  K.class_group()

21.  You can use reference manual or quick search in www.sage.orgwww.sage.org  use help() in Sage  function??: can see source code of function ◦ factor??  Tab completion: Type obj followed by tab to see all completions of obj. ◦ fac+tab

22. Thank you!