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7.5.2  Minimum polynomial of an algebraic number

The minimal polynomial of an algebraic number is the monic polynomial of smallest degree with integer coefficents which has the algebraic number as a root.

The pmin command finds the minimum polynomial of an algebraic number.

Examples

pmin(sqrt(2)+sqrt(3))
     
▯ 1,0,−10,0,1▯           
pmin(sqrt(2)+sqrt(3),x)
     
x4−10 x2+1           

Note that (√2+√3)2=5+2√6 and so ((√2+√3)2−5)2=24, which can be rewritten as (√2+√3)4−10 (√2+√3)2+1=0.

pmin(sqrt(2)+i*sqrt(3))
     
▯ 1,0,2,0,25▯           
pmin(sqrt(2)+i*sqrt(3),z)
     
z4+2 z2+25           
pmin(sqrt(2)+2*i)
     
▯ 1,0,4,0,36 ▯           
pmin(sqrt(2)+2*i,z)
     
z4+4 z2+36           

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