Are there any linkage or transformation between some period transcendental numbers and algebraic irrational numbers?

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Are there any linkage or transformation by combination of integral and algebraic function like in the definition of period number between some period transcendental numbers and algebraic irrational numbers,especially these irrational algebraic number exceeding degree 2? The relation or linkage between transcendental constants like $ \pi,e,\dots$ and algebraic irrational numbers,especially these irrational algebraic number exceeding degree 2 is much interesting to me.