In Milnor and Stasheff's book "Characteristic Classes", problem 7b, they ask us to show that the cohomology ring $H^* (G_n (\mathbb{R}^{n+k}, \mathbb{Z}_2)$ is generated by the Stiefel-Whitney classes $w_1, \ldots , w_n$ of the tautological bundle $\gamma^n$ of the infinite Grassmannian $G_n$, and the dual classes $\bar{w_1}, \ldots , \bar{w_k}$, and we take the quotient of $$ (1+w_1 + \ldots + w_n)(1 + \bar{w}_1 + \ldots + \bar{w}_k)= 1.$$
The dual class $\bar{w}_k$ is defined to be the multiplicative inverse of $w_k$ in the cohomology group $H^k$.
Now before I tackle this problem I'm just wondering what is the correct algebraic formulation of the cohomology ring. Should it be $$ \frac{\mathbb{Z}_2 [w_1 , \ldots , w_n]}{\langle w\bar{w}-1 \rangle} $$ where $w = 1 + w_1 + \ldots + w_n$, or rather should it be written as $$ \frac{\mathbb{Z}_2 [w_1 , \ldots , w_n, \bar{w}_1 , \ldots , \bar{w}_k]}{\langle w\bar{w}-1 \rangle} ?$$
Or perhaps, both formulations are completely wrong and I'm misunderstanding what's stated in the problem?
Either way, I'm tempted to go with the first because I know each inverse SW-class can be written as a polynomial expression of the usual SW-classes of $\gamma^n$, but I'm wondering if I'm actually losing information by omitting the inverses from the first expression. But then the second one becomes heavy and cumbersome to write, and I'm lazy and hope for nice, simple formulas.
EDIT: I've found a third and definitely correct formulation: $$ \frac{\mathbb{Z}_2 [ w_1 , \ldots , w_n ]}{ \langle \bar{w}_{k+1} , \ldots \bar{w}_{n+k} \rangle}. $$
Now the next question is, is this equivalent to what I have written above? I'm guessing the ideal I have in my two attempts can split into those $n$ inverse SW-classes.
