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How to Join the Brotherhood of Steel
The Brotherhood of Steel is the most iconic group in the Fallout universe, striding around in Power Armor and protecting the Commonwealth. They are the do-gooders, and youll probably want to join them as soon as possible. Not least because once you do you get handed a tasty customised laser rifle.

Firstly, make sure you have started The First Step quest, which marks the Corvega Assembly Plant on your map. You dont need to complete the quest at this stage. Head over to the plant, and circle around the building. Its around here youll pick up the Military Frequency AF95 radio signal. If you havent started The First Step, head over to here on your world map:
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Youll hear a distress call asking for help, which will mark the Cambridge Police Station on your map. This is the Fire Support quest.

Head over to the location. As you arrive youll see Brother Danse in power armor fighting off a whole bunch of ghouls and a super mutant. Help him out (theres ammo here if you need it) and when he battle is over talk to him.

This will kick off the Call To Arms quest, and its completing this that will result in an invite to join the Brotherhood. Go inside the police station and load up on ammo and anything else you can loot. Check out the Motor Pool for some extra ammo too. You can read Danses diary on the computer here if you like. When youre ready, talk to Danse.
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Youre now going to get a Deep Range Transmitter from ArcJet Systems. Danse knows the way, so follow him out the station and along the shoreline where youll run into a few ghouls. Danse tells you his story on the way and youll be there in no time.

Inside ArcJet, have a look around, then arm a decent weapon before opening the twin doors. As they open youll both be rushed by synths. These synthetic humans arent too tough, but they can become overwhelming in high numbers so dont get cocky.

The good thing here is that even if youre not the toughest character at this point Brother Danse will do a lot of the work. Hes rock hard. Both of you work your way down into the depths of the building to fix the elevator, fighting handfuls of synths on the way. At the bottom youll have to fix the power. Theres a fusion core here, so make sure you pick that up before cracking the computer terminal.

As soon as the power is back on a shit-tonne of synths will drop down into the main room. Danse stays there and fights them, but youre better off using the corridor for cover and attacking from the corner. If you step into the main room youll likely get cut to ribbons by laser fire. Theres a lot of bad guys.

Once theyre all dead, ride the elevator up to the control room and kill the rest of the synths. Search the bodies to find the Deep Range Transmitter, then take the next elevator up and outside.

How to get Righteous Authority
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Talk to Danse to compete the task and its at this point he will gift you his modified laser rifle, Righteous Authority.

He will also now ask you to join The Brotherhood of Steel, which initiates the Sempa Invicta quest. Say yes and meet him back at the Cambridge Police Station. Congratulations, you are now a member of the Brotherhood of Steel!