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Re: Factor[1+x^4]
*To*: mathgroup at smc.vnet.net
*Subject*: [mg26889] Re: Factor[1+x^4]
*From*: paradaxiom at my-deja.com
*Date*: Fri, 26 Jan 2001 01:27:32 -0500 (EST)
*Sender*: owner-wri-mathgroup at wolfram.com
By default Factor[] factors over integers. If you want Factor[]
to factor into Sqrt[2] and I you have to give this to the
Extension option:
In[1]:= f = Factor[1 + z^4, Extension -> {I, Sqrt[2]}]
Out[1]= (1/4)*(Sqrt[2] - (1 + I)*z)*
(Sqrt[2] - (1 - I)*z)*
(Sqrt[2] + (1 - I)*z)*
(Sqrt[2] + (1 + I)*z)
In[2]:= Expand[f]
Out[2]= 1 + z^4
Of course, in general you don't know what Extension to pick beforehand.
So one thing you can do is the following:
In[1]:= poly = 1 + z^4
Out[1]= 1 + z^4
The find the extension you want by looking at the roots of the
polynomial:
In[2]:= extension[poly_, z_] :=
ComplexExpand[
Roots[poly == 0, z] /. {_ == a_ -> a} /. {Or -> List}
]
In[3]:= extension[poly, z]
Out[3]= {(1 + I)/Sqrt[2],
-((1 - I)/Sqrt[2]),
-((1 + I)/Sqrt[2]),
(1 - I)/Sqrt[2]}
In[4]:= Factor[poly, Extension -> extension[poly, z]]
Out[4]= (1/4)*(Sqrt[2] - (1 + I)*z)*
(Sqrt[2] - (1 - I)*z)*
(Sqrt[2] + (1 - I)*z)*
(Sqrt[2] + (1 + I)*z)
I hope this helps a bit. Perhaps someone knows a better way to
go about this?
//Marten
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