Polynomial ring of level-one modular forms ($M_{\ast }(S{L}_2(\mathbb{Z}))=\mathbb{C}[E_4,E_6]$)

Subtract the constant term of a modular form times a same-weight monomial in $E_4,E_6$. The remainder is a cusp form, hence divisible by the modular discriminant. Induction on weight and $1728\Delta =E_4^3-E_6^2$ express every form as a weighted homogeneous polynomial in these two Eisenstein series. The number of monomials of a given weight equals the dimension of level-one modular forms, so they form a basis and the graded ring has no polynomial relation.

Let $P_w(E_4,E_6)$ be weighted homogeneous of modular weight $w$, and suppose ${\sum }_w{P}_w(z)=0$ as a holomorphic function. Apply the modular group matrices $\left(\begin{array}{cc}1& 0\\ n& 1\end{array}\right)$. At a fixed point $z$ in the complex upper half-plane, this gives ${\sum }_w(nz+1{)}^w{P}_w(z)=0$ for every integer $n$. A polynomial with infinitely many roots vanishes identically, so every $P_w(z)=0$. Within one modular weight, the monomials reduce to powers of $E_4^3/E_6^2$, after taking out a common factor. This ratio is nonconstant, as its Fourier expansion begins $1+1728q+\cdots$, so its powers are linearly independent. This proves algebraic independence even when the generators are regarded as analytic functions rather than elements of a formal graded ring.

A level-one modular form with rational Fourier coefficients is a weighted homogeneous polynomial with rational coefficients in the normalized Eisenstein series $E_4,E_6$. Subtract its constant coefficient times a same-weight monomial. The remainder is a cusp form; divide it by ${\Delta }_0=(E_4^3-E_6^2)/1728=q+O(q^2)$. The valence formula for the modular group shows that ${\Delta }_0$ has no interior zeros, so this quotient is holomorphic and has rational coefficients. Induction on weight proves the assertion. Euler's Bernoulli number formula for even zeta values then gives the corresponding rational weighted polynomial in the unnormalized lattice Eisenstein sums $G_4,G_6$.