GaussianIntegers

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GaussianIntegers

is an option for FactorInteger, PrimeQ, Factor, and related functions that specifies whether factorization should be done over Gaussian integers.

Details

With GaussianIntegers->False, factorization is done over the ordinary ring of integers .
With GaussianIntegers->True, factorization is done over the ring of integers with i adjoined .
The Gaussian primes used when GaussianIntegers->True are chosen to have both real and imaginary parts positive.
The first entry in the list given by FactorInteger with GaussianIntegers->True may be -1 or -I.

Examples

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Basic Examples (1)Summary of the most common use cases

Factor a polynomial over :

In[1]:=1

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https://wolfram.com/xid/0ywh8bcjdf3c-b2m6fm

Out[1]=1

Factor an integer over :

In[2]:=2

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https://wolfram.com/xid/0ywh8bcjdf3c-crn5kc

Out[2]=2

Scope (3)Survey of the scope of standard use cases

By default polynomial factorization is performed over the rationals:

In[1]:=1

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https://wolfram.com/xid/0ywh8bcjdf3c-4ql3h

Out[1]=1

This specifies that the factorization should be done over :

In[2]:=2

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https://wolfram.com/xid/0ywh8bcjdf3c-ffoavo

Out[2]=2

By default integer factorization is performed over the integers:

In[1]:=1

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https://wolfram.com/xid/0ywh8bcjdf3c-ic6qe2

Out[1]=1

This specifies that the factorization should be done over the Gaussian integers:

In[2]:=2

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https://wolfram.com/xid/0ywh8bcjdf3c-l50iga

Out[2]=2

A number prime over the integers may not be prime over the Gaussian integers:

In[1]:=1

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https://wolfram.com/xid/0ywh8bcjdf3c-jqwj4k

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ywh8bcjdf3c-krrkvs

Out[2]=2

Properties & Relations (1)Properties of the function, and connections to other functions

For Factor, GaussianIntegers->True is equivalent to Extension->I:

In[1]:=1

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https://wolfram.com/xid/0ywh8bcjdf3c-btgb1f

Out[1]=1

In[2]:=2

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https://wolfram.com/xid/0ywh8bcjdf3c-bs2ioe

Out[2]=2

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