-- In M2 things with a -- in front of them are comments, i.e. are ignored -- by the software --sometimes we want to resart to clear variables restart --Create a polynomial ring over the rationals R=QQ[x] --We can also do several variables R=QQ[x,y,z] R=QQ[w_1..w_6] --Use coefficents from a finite feild R=ZZ/32749[x] R=ZZ/32749[v_1..v_4] isPrime 977 R=ZZ/977[x] --We can define ideals R=ZZ/32749[v_1..v_4] I=ideal(v_1^2-6*v_2,v_3*v_1-45) --and check if they are prime isPrime I --back to one variable R=QQ[x] I=ideal(x^2-23*x+47) isPrime I --get the first (and only) generator of I f=I_0 --we can try to factor it, and see it is irriducible (which we already knew since the ideal is prime) factor(f) --We can define a quotient ring S=R/I --we can do products of things in S h=(x^7+24)*(x^2-x-34) --since S is a feild we can also find inverses (note we need the two slashes) g=1//h g*h --we could also do the following, sing S is a feild (x^2-5)//(x+4) --One should be careful about reusing names too much in different rings... this can cuase errors, we will restart to avoid this restart --We can also find gcd's R=QQ[x] f=x^2-23*x-2 g=23*x^3-34 gcd(f,g) --since these polynomials have gcd 1 then an ideal containing both must be the whole ring --we can check equality of rings and ideals and ideals and ideals with == ideal(f,g)==R h=f*(x+5) --now we will get a polynomial other than 1 b=gcd(h,f) --note that the gcd(h,f) will define the same ideal as f and h ideal(f,h)==ideal(b) --We can also define Quotient rings in one line S=QQ[z,y]/(z^2-y^2,z-1) S=QQ[x]/(x^4-23*x^3-34) describe S