AOL will begin trading today on the New York Stock Exchange as an independent company for the first time in eight years. Today, the two companies will go their separate ways eight years after their $160 billion merger in 2001. Each will be a smaller version of what they were a decade ago. AOL, once the largest dial-up internet service provider in the US, will be preparing to do battle with Google, Yahoo! and Microsoft in the $29 billion US online advertising market.