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NumberTheory`PrimitiveElement`
This package provides a function for computing primitive elements of multiple algebraic extensions of rationals.
Given algebraic numbers  ,  ,  we can always find a single algebraic number  such that each of  ,  ,  can be expressed as a polynomial in  with rational coefficients. The number  is called a primitive element of the field extension  ( ,  ,  )/ . In other words, an algebraic number  is a primitive element of  ( ,  ,  )/ iff  ( ,  ,  ) =  ( ).
The function PrimitiveElement takes a variable  and a list of algebraic numbers  ,  ,  and returns a primitive element  of  ( ,  ,  )/ , and a list of polynomials  ( ),  ,  ( ) such that  =  ( ) for all 
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The primitive element of an algebraic extension of the rationals.
This loads the package.
In[1]:= <<NumberTheory`PrimitiveElement`
Here is a primitive element of  (Sqrt[2], Sqrt[3])/ , and a list of polynomials showing how to represent Sqrt[2] and Sqrt[3]
in terms of the primitive element.
In[2]:= PrimitiveElement[z, {Sqrt[2], Sqrt[3]}]
Out[2]= 
This checks that the computed polynomials evaluated at the primitive element give Sqrt[2] and Sqrt[3].
In[3]:= RootReduce[%[[2]] /. z -> %[[1]]]
Out[3]= 
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