curious about multiple defined symbols.

Let's say I have a header file, that is included by two different .cpp files.

//test.h
#ifndef TEST_H
#define TEST_H

int g_int;
#endif

//test1.cpp
#include "test.h"
extern int g_int;
...

//test2.cpp
#include "test.h"
extern int g_int;

I am getting the link error LNK2005: "int g_int" (?g_int@@3HA) already defined in test1.obj

Now I thought that declaring them as extern in the .cpp files would tell the compiler that the g_int variable was already declared externally.  I know that I can move the g_int declaration to the .cpp file and it will work fine.  I was just curious if there was any way at all to have a global variable in a header file.