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Get the Unifi UAP Lite - it does power over ethernet so you can put the POE injector (included) at one end and the UAP Lite at the other. You Wi-Fi will be awesome. Simples.
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What you are looking for is a 'wireless access point', not wifi booster/range extender/repeater which are about extending well, wirelessly, rather than via Ethernet. (Although the terms seem to get confused, especially with powerline networking... )
Very cheapest way of doing it is to use an old wifi router and put it into 'bridge'/'client' /'AP' mode. This may not be possible on BT home routers though. Should be easy to find an old Netgear/Linksys/TP-Link etc for a couple of quid though especially if you are OK with older wireless N.
If not something like this does wireless AC: https://www.ebuyer.com/748436-tp-link-ap300-ac1200-wireless-gigabit-access-point-ap300 for about fifty quid.
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chez_jay
@skinny
I need a wifi booster than does so via an ethernet cable.
So I have an ethernet cable that runs a long way (brick building) and I want to then plug that into a device to give wifi.
What are my options?