The singleton set α is algebraically independent over K if and only if the element α is transcendental over K.

A subset S⊂L is algebraically independent over K if every element of S is transcendental over K and over each of the extension fields over K generated by the remaining elements of S.

1999, David Mumford, The Red Book of Varieties and Schemes: Includes the Michigan Lectures, Springer, Lecture Notes in Mathematics 1358, 2nd Edition, Expanded, page 40,
If the statement is false, there are n elements x_1,…,x_n in R such that their images ◌̅x_i in R/P are algebraically independent. Let 0 ne p∈P. Then p,x_1,…,x_n cannot be algebraically independent over k, so there is a polynomial P(Y,X_,…,X_n) over k such that P(p,x_,…,x_n)=0.

Setting y_i=y_i,1 (where 1 denotes the identity of the semigroup T) we obtain a σ-algebraically independent over R set y_i|i∈I such that S=R(y_i)_(i∈I).

If α ne 0,1 is algebraic and β is an algebraic irrational of degree d>2, then
αᵝ,…,α
are algebraically independent.

Algebraically independent

Meaning

Adj