First of all, Alaska was sold to the United States by the then Russian Empire (Russia), and now Alaska belongs to the United States without any objection. Secondly, Alaska is not a sphere of influence or colony controlled by the United States, but a solid land and one of the 50 States in the United States. So Putin has no condition to control Alaska.
I think the current Russian government is even more afraid to have any thoughts about Alaska, because Alaska belongs to the United States, the most powerful country in the world. Who dares to grab food from the tiger's mouth? Although the ice and snow in Alaska were worthless, a large number of gold mines, oil and other minerals were discovered in Alaska one after another and became a treasure trove of geomantic omen. Although Russia regrets its greedy eyes, it can only do so and will definitely not take back Alaska.
Moreover, first of all, the United States is beyond Russia's ability to cope. What is more troublesome is that Russia's current predicament is the military containment and economic sanctions of western countries led by the United States. It is not easy for Putin's government to keep Russia's existing territory and interests. Besides, Russia doesn't need Alaska. What Russia lacks is population, regardless of area.
The resources in Siberia, Russia are seven or eight times richer than those in Alaska, and the environment is similar to that in Alaska. Alaska is destined to be just a page in Russian history, and it should have been exposed long ago. Now we should cherish the present resources more.