Affine n - spaces over a field $K$ is the cartesian product of the field $K$ with itself $n$ time and it is denoted by $\mathbb A^n(K)$.
$X$ is a subset of Afine n - spaces $\mathbb A^n(K)$ is called algebraic if there exist an ideal I of $K[x_1,X_2, \cdots , X_n]$ such that $V(I) = X$, where $V(I) = \{f \in I : f(a_1, a_2 , \cdots, a_n) = 0 \ \ \text{for all} \ (a_1, \cdots , a_n) \in X \}$
Which of the following subsets are algebraic or not ?
$A = \{ (x,y) \in \mathbb A^2(\mathbb R) : y = \sin x \}$
$B=\{ (\cos t, \sin t) \in \mathbb A^2 (\mathbb R) : t \in \mathbb R \}$
$C=\{ (z,w) \in \mathbb A^2(\mathbb C) : |z|^2 + |w|^2 = 1 \}$
$D =\{ ( \cos t , \sin t, t) \in \mathbb A^3(\mathbb R) : t \in \mathbb R \}$
I have an idea about (2), let $I = < x^2 + y^2 -1>$, then $B = I$, I think $(1), (3), (4)$ are not algebraic , but i do not know how to prove .
I would be thankful if someone help me to prove. Thank you.
For the first one, suppose a polynomial f(x,y) vanishes on the locus y=sin(x). Then f(x,0) is a polynomial in one variable vanishing at each multiple of 2pi. What does that tell us about f(x,0)?