# ag.algebraic geometry – direct sum and algebraically independent of monomials in the polynomial algebra

Let $$A=K[X_1, X_2,dots, X_8]$$ be the polynomial algebra in 8 variables and $$K$$ is an algebraically closed field of characteristic zero. Let $$a_1=X_3X_4, a_2=X_5X_6, a_3=X_7X_8$$, and $$a_4=X_7X_4X_6$$.

I have two questions:

1. prove that the sum $$Sigma_{n_1, dots, n_6geq 0}KX_1^{n_1}X_2^{n_2}a_1^{n_3}a_2^{n_4}a_3^{n_5}a_4^{n_6}$$ is dirct sum if and only if the monomials $$X_1, X_2, a_1, a_2, a_3, a_4$$ are algebraically independent in the polynomial algebra $$A$$.

2. prove that the monomials $$X_1, X_2, a_1, a_2, a_3, a_4$$ are algebraically independent in the polynomial algebra $$A$$.

Thank you for any help