Are my following conjectures formulated linguistically and mathematically correct?
1.) $K$ be a field, $O$ a superfield of $K$, and $v_1,...,v_n$ be algebraically dependent over $K$. $v_1,...,v_2$ are algebraically dependent over $O$ then.
2.) $K$ be a field, $U$ a subfield of $K$, and $v_1,...,v_n$ be algebraically independent over $K$. $v_1,...,v_2$ are algebraically independent over $U$ then.
3.) $K$ be a field, $\overline{K}$ the algebraic closure of $K$, and $v_1,...,v_n$ be algebraically independent over $K$. $v_1,...,v_n$ are algebraically independent over $\overline{K}$ then.
Are these conjectures true?
Is this obvious and need not be proven?
Do you have some literature references?
Question 1: Alg. dependence over $K$ implies alg. dependence over a field extension of $K$. Yes, indeed. If an equation holds over $K$, it holds over the field extension.
Question 2: This is the contraposition of question 1: Alg. independence of a field extension of $K$ implies alg. independence over $K$.
Question 3: Your link points in the right direction.