I have a question about the following problem from the Grobner bases class:
Given the polynomial ring $\mathbb{Q}[x, y, z, t]$ and $A=\langle f_{1},f_{2},f_{3},f_{4} \rangle$ where $f_{1}=x-2y+z+t,\;f_{2}=x+y+3z+t,\;f_{3}=2x-y-z-t,\;f_{4}=2x+2y+z+t.$
Q: Find a better generating set for $A$, and determine the basis for $\mathbb{Q}[x, y, z, t]/A$ as a vector space over $\mathbb{Q}$.
My work:
By row reducing the system of polynomials $f_{1},\;f_{2},\;f_{3},\;f_{4}$. I found the better generating set for $A$ is $\langle x,y,z,t \rangle$. Thus, the coset representatives for the quotient ring $\mathbb{Q}[x, y, z, t]/A$ is $c\;+\;\langle x,y,z,t \rangle$, where $c$ is a constant polynomial.
My question:
Isn't the basis for $\mathbb{Q}[x, y, z, t]/A$ is just $\{1+\langle x,y,z,t \rangle\}$? I am a little bit unsure about this.