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no you can not
You can't play minecraft alpha 1.2.6. You can currently play minecraft beta 1.8 if you buy the game for $21.95 and then play it in the browser or download it to your computer and play it in an application. Alpha 1.2.6 is long gone, and it is not able to be found anywhere, as far as I know.
It is free to make an account on the Minecraft website, but you have to pay to play the game.
Minecraft is a onetime Purchase not every game is a pay to play...pay to play meaning you pay per month to play.
Minecraft is all about the account you made, you can log into that account on any computer, you are not forced to play it only on the computer that you purchased it on.