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**NonCommAlg** (a): let

(c) this part is nice! suppose $\displaystyle x^n - t = r(x) s(x),$ for some $\displaystyle r(x)=\sum_{i=0}^k r_ix^i, \ s(x)=\sum_{i=0}^m s_i x^i, \ k, m > 0, \ r_i, s_i \in F[[t]].$ then $\displaystyle r_0s_0=-t$ and

hence, without loss of generality, we may assume that $\displaystyle r_0=tu, \ s_0=-u^{-1},$ for some invertible element $\displaystyle u \in F[[t]].$ to complete the proof of

(c), use induction to show that, in $\displaystyle F[[t]],$ we have $\displaystyle t \mid r_i,$ for all $\displaystyle i,$ which is the contradiction we need.