examples of minimal polynomials
Note that is algebraic over the fields and . The minimal polynomials for over these fields are and , respectively. Note that is irreducible over by using Eisenstein’s criterion and Gauss’s lemma (http://planetmath.org/GausssLemmaII) (see this entry (http://planetmath.org/AlternativeProofThatSqrt2IsIrrational) for more details), and is irreducible over since it is a quadratic polynomial and neither of its roots ( and ) are in .
A common method for constructing minimal polynomials for numbers that are expressible over is “backwards ”: The number can be set equal to , and the equation can be algebraically manipulated until a monic polynomial in is equal to 0. Finally, if the monic polynomial is not irreducible, then it can be factored into irreducible polynomials , and the original number will be a root of one of these. A very example is :
This method will be further demonstrated with three more examples: One for , one for where is a fifth root of unity, and one for .
Since is a quadratic and has no roots in , it is irreducible over . Thus, it is the minimal polynomial over for .
On the other hand, factors over as . Since is not a root of , it must be a root of . Moreover, this polynomial must be irreducible. This fact can be proven in the following manner: Let be the minimal polynomial for over . Since , . (Here denotes the Euler totient function.) Since divides and they have the same degree, it follows that .
It turns out that is irreducible over . (This can be proven in a manner as above. Note that .) Thus, it is the minimal polynomial over for .
|Title||examples of minimal polynomials|
|Date of creation||2013-03-22 16:55:18|
|Last modified on||2013-03-22 16:55:18|
|Last modified by||Wkbj79 (1863)|