The problem is that the Realtek ethernet chip is not properly initialized. The driver r8169 is selected automatically by Linux and doesn't work.
The current solution is to downgrade the driver to r8168. Follow the steps below, the `uname -r` substitutes the linux kernel name in the path:
1) download the driver r8168-8.015.00.tar.bz2 and burn it on a cd for safe keeping and future use.
2) log in as root with the password from the nucore users manual
3) use the file viewer to extract the driver to the /root directory or type tar -vjxf r8168-8.015.00.tar.bz2 in the /root directory
4) cd r8168-8.015.00
5) rmmod r8169
6) mv /lib/modules/`uname -r`/kernel/drivers/net/r8169.ko ~/r8169.ko.backup
7) make clean modules

9) depmod -a
10) insmod ./src/r8168.ko .
11) mv /initrd.img ~/initrd.img.backup
12) mkinitramfs -o /boot/initrd.img-`uname -r` `uname -r`
13) echo "r8168" >> /etc/modules
14) echo -e "nblacklist r8169n" >> /etc/modprobe.d/blacklist
Reboot linux to test the repair.