The one-field $N=2$ Landau–Ginzburg Jacobian algebra has dimension k+1

By the derivative definition in [F1], $W^{\mathrm{\prime }}(X)=\lambda X^{k+1}$. Since $\lambda \mathrm{\ne }0$ is a unit by [F2], the ideals $(W^{\mathrm{\prime }})$ and $(X^{k+1})$ are equal: each generator is a unit multiple of the other. Thus the quotient in [F1] is $\mathbb{C}[X]\mathrm{/}(X^{k+1})$.

Define scalar multiplication on $\mathbb{C}[X]$ by $a\cdot {\sum }_j{b}_j{X}^j={\sum }_j(a{b}_j)X^j$. Its addition is coefficientwise; the additive-group laws follow coefficientwise from the ring structure [F4], and the scalar distributivity, associativity, and identity axioms follow coefficientwise from the field laws [F2] and the polynomial operations [F3]. Hence this makes $\mathbb{C}[X]$ a complex vector space in the sense of [F6].

Let $I=(X^{k+1})$. By [F5], $I=\{q{X}^{k+1}:q\in \mathbb{C}[X]\}$. It contains zero; if $q{X}^{k+1},q^{\mathrm{\prime }}{X}^{k+1}\in I$, their sum is $(q+q^{\mathrm{\prime }})X^{k+1}\in I$; and for $a\in \mathbb{C}$, $a\cdot (q{X}^{k+1})=(a\cdot q)X^{k+1}\in I$. Thus $I$ is a linear subspace of $\mathbb{C}[X]$ by [F8].

By [F7] and step 1.3, the additive cosets $\mathbb{C}[X]\mathrm{/}I$ in the quotient-ring construction of [F1] carry the quotient vector-space operations $[p]+[q]=[p+q]$ and $a[p]=[a\cdot p]$. Therefore the quotient ring in step 1.1 has this specified complex vector-space structure.

For a finitely supported polynomial $p(X)={\sum }_{j=0}^N{a}_j{X}^j$, extend its coefficient sequence by $a_j=0$ for every $j>N$, and write $p=q{X}^{k+1}+r$, where $q={\sum }_{j=k+1}^N{a}_j{X}^{j-k-1}$ (the zero polynomial if $N\le k$) and $r={\sum }_{j=0}^{\min (N,k)}{a}_j{X}^j$. Then $[p]=[r]={\sum }_{j=0}^k{a}_j[X^j]$ in the quotient vector space, so every quotient vector is a finite linear combination of $B=\{[1],[X],\dots ,[X^k]\}$. The span of $B$ is a linear subspace containing $B$ by the span definition in [F9], so it contains every such combination; hence $B$ spans the quotient.

Suppose ${\sum }_{j=0}^k{a}_j[X^j]=0$. By quotient-coset equality this means $r={\sum }_{j=0}^k{a}_j{X}^j\in I$. By [F5], $r=q{X}^{k+1}$ for some polynomial $q$. Convolution multiplication in [F3] gives zero coefficients in degrees $0,\dots ,k$ for $q{X}^{k+1}$, whereas those coefficients of $r$ are $a_0,\dots ,a_k$; hence each $a_j=0$. The finite-list criterion [F9] gives linear independence of the displayed classes.

Let $B=\{[1],[X],\dots ,[X^k]\}$. The map $j\mapsto [X^j]$ from $\{0,1,\dots ,k\}$ onto $B$ is injective: if $[X^i]=[X^j]$ for $i\mathrm{\ne }j$, their difference gives a nontrivial vanishing linear combination, contradicting step 3.2. To check independence of the subset $B$, take any injective finite list from $B$ and assign its coefficients to the distinct exponent indices of its entries, with zero coefficients at the unused indices; step 3.2 forces all coefficients to vanish. Thus $B$ is linearly independent as a subset. Together with spanning from step 3.1, [F9] makes $B$ a basis with $k+1$ elements, so the dimension definition in [F9] gives ${\dim }_{\mathbb{C}}\mathrm{Jac}(W)=k+1$. This is an algebraic quotient calculation only; no infrared-CFT identification is part of this theorem.