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Is your device, that shows the authentication error, an Android handheld device or phone or tablet? If not, please tell me what it is. Thanks.
Thanks for the info in response. I'm considering my answer and preparing advice...
The problem may be one of several things.
1: your router.... do other devices connect to it OK?
2: your Kindle Fire... how old is it? has it worked with a WiFi connection before this, in your experience?
3: are you near enough to the router that the signal from it is assuredly strong?
4: are you using the Kindle Fire in 'Landscape' mode? it has been reported that can obscure needed information for the connection, that becomes visible when the Kindle is rotated into Portrait mode.
Good. That makes it sound as if the Kindle wireless/WiFi ability has failed.
There are some things you can do:
1. if it's a matter of minor corruption of the Kindle's operating software, doing a reset back to factory conditions will erase everything stored on the Kindle - notes, bookmarks, books donwloaded onto it - but may restore WiFi connection. Back up notes and bookmarks, of course, first... if possible.
2. if it's a matter of the Kindle being 'broken', it can be sent to Amazon for repair - at cost to you, since it is out of warranty.
3. if it might just be something else... a good test first is to take it to another WiFi source, briefly, and see if it connects there, even briefly, after authenticating. If it does, then (1) and (2) are not the problem.
(I'm sorry - I had trouble seeing your response at first. I'm preparing an answer that you can read later.)
Swipe down from the top of the screen and
tap on Settings.
Tap Device Options, and then
tap on Reset To Factory Defaults.
Tap Reset, to confirm.
If you need to back up data off the Kindle first, ask me about how to do so before trying the reset. It will, to repeat, wipe all data off the Kindle device, period.