You may opt any of the followings:-
1]Right click on LAN icon>status>properties>uncheck 'Show icon in
notification area when connected'/Notify me when this connection has
limited or no connectivity.
2]You can hide the icon by right-clicking the taskbar, click on
Customize,then find the local area connection in the list, click on it
and under behavior, click on "Always hide"
or,
3]If you don't use it then you can also Right click on My Network
Places, Properties, right click on Local Area Connection and select Disable.
regards,
ssg MS-MVP