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# free module changes dimensions after base change

Hi, I have a free module of rank 1 over the integers

sage: M

Free module of degree 1 and rank 1 over Integer Ring
Echelon basis matrix:
[2]


However, when I base change it to the field with two elements, it changes dimension!

sage: M.change_ring(GF(2))

Vector space of degree 1 and dimension 0 over Finite Field of size 2
Basis matrix:
[]


What's going on?

P.S.: What does Sage mean by "degree"?

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## 1 answer

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The degree means the dimension of the ambient module. For example, the vector space spanned by (2,0,2) inside R^3 is one dimensional but is of degree 3.

In your case, you have a module over the integers, perhaps spanned by (2) inside of a rank one free Z-module. When you reduce mod 2, the spanning vector becomes (0), so the resulting module over GF(2) is zero dimensional.

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## Comments

So, change_ring applies to the ambient module and not the module itself? That's messed up! Is there a way to base change the module itself then?

( 2018-05-21 18:15:27 -0600 )edit

Note that this behavior of change_ring is described it its documentation. Anyway, mathematically, what are you trying to accomplish?

( 2018-05-22 09:49:09 -0600 )edit

I have a homomorphism between two free modules and what to base change it to a finite field.

( 2018-05-23 09:07:45 -0600 )edit

Mathematically, if you have the times 2 map between two free modules and you change to GF(2), that map should become zero. That is what Sage's behavior is modeling.

( 2018-05-23 09:56:10 -0600 )edit

The map that I want to base change is not multiplication by two. In fact, trying to base change the map in question throws an exception, precisely because the dimension of the codomain changes.

( 2018-05-24 11:47:26 -0600 )edit

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Asked: 2018-05-21 15:27:29 -0600

Seen: 63 times

Last updated: May 21 '18