If you have booked your flight it will automatically send the confirmation mail to your Gmail, Yahoo account according to the mail you mentioned while booking, if in case you have booked through American Airlines and didn't confirmation mail even after paying the amount there might be chances that the airline is still processing the mail. American Airlines normally takes up to 4 hours to send the email confirmation to the passenger after booking.
In case you don't receive your email confirmation even after booking then try these things -
Make sure that your payment was successful. As sometimes due to bad network and bank issues the amount is refunded back to your account.
Call the Airline to know about the status of your ticket after verifying your details.
Check your Spam folder. Sometimes your mail automatically filters a few emails to the Spam folder. If you find the mail in the Spam folder drag and shift it to the Primary tab.
The actual reason why you did not receive an immediate email is that the airlines do not actually book the ticket immediately. There are hundreds of people queued up in the row for booking as you. So there might be chances of delay in the verification process of the person who booked ahead of you. However, If you need immediate verification or in case you have booked a flight through American Airlines did not get a confirmation email even after 4 hours then you should contact the Airlines or the agencies through which you booked directly.
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I would like to note that I have never had any complaints about American Airlines, everything is always at the highest level with ticketing and service. It's a shame that many other airlines do the same thing, such as not being able to check seat availability (for example, I want a window seat) and ticket price without buying a ticket. I'd rather do that than just get a random seat. I'm a writer who works for one of the best services https://writemyessay.nyc, and during a flight when I look out the window I get new ideas. I understand that a lot of people want to go through the clearance process, but sometimes it's not worth it. I'd rather be able to see if the seats I want are available and if so, find out if the price is reasonable.