Compute the (multiplicative) inverse of $4x+3$ in the field $\frac {\Bbb F_{11}[x]}{\langle x^2+1 \rangle}$?

abstract-algebra ring-theory proof-verification quotient-spaces

Solution 1:

Here's an alternative approach, doable since the vector space here is only two-dimensional. Multiplication by $4x+3$ corresponds to the matrix $\begin{bmatrix}3&7\\4&3\end{bmatrix}$ using the basis $\{1,x\}$. This matrix has inverse $\frac{1}{-19}\begin{bmatrix}3&-7\\-4&3\end{bmatrix}\equiv\frac{1}{3}\begin{bmatrix}3&-7\\-4&3\end{bmatrix}\equiv4\begin{bmatrix}3&4\\7&3\end{bmatrix}\equiv\begin{bmatrix}1&5\\6&1\end{bmatrix}$, which corresponds to multiplication by $1+6x$ (upon inspecting the first column).

Solution 2:

Yes, that is very good. And you can check that $(4x+3)(6x+1) = 24x^2 + 22x + 3 = 2x^2 + 3 = 2(-1) + 3 = 1$ as required.

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