# Solving an equation in a field.

I need to know the way to solve equations like this: $$(x^2+1)f(x) = 1 \pmod{x^3+1}$$ over a field $F_{3}[x]$.

Thanks in advance for any help.

• Hint: as a start, observe that $x^3+1 = (x+1)^3$ in $\mathbb F_3[x]$. – Dilip Sarwate Nov 17 '13 at 14:45
• Do you know how to apply the Euclidean algorithm? – Calvin Lin Nov 17 '13 at 14:56
• Nitpick: $F_3[x]$ is not a field but a polynomial ring :-) – Jyrki Lahtonen Nov 17 '13 at 14:56
• But to give you hint (I was away watching a fantastic darts match): Any residue class modulo $x^3+1$ has a unique representative of the form $a(x)=a_0+a_1x+a_2x^2$ with $a_0,a_1,a_2\in F_3$. Your task is to find the values of those constants in such a way that $a(x)(x^2+1)$ leaves remainder $1$ modulo $x^3+1$. You can brute force this either by testing all 27 combinations, or you can replace it with a linear system of three equations in the three unknowns, or you can use Calvin Lin's hint. – Jyrki Lahtonen Nov 17 '13 at 16:01
• Thanks very much for your valuable hint Jyrki...It helped me a lot and I found the answer as : x^2+x+2. – Ali Nov 17 '13 at 19:24

## 1 Answer

Hint: Any residue class modulo $x^3+1$ has a unique representative of the form $a(x)=a_0+a_1x+a_2x^2$ with $a_0,a_1,a_2\in F_3$. Your task is to find the values of those constants in such a way that $a(x)(x^2+1)$ leaves remainder 1 modulo $x^3+1.$ You can brute force this either by testing all 27 combinations, or you can replace it with a linear system of three equations in the three unknowns, or you can use Calvin Lin's hint.

• Moved the hint (that judging from OP's reaction) from comments to a CW-answer, as the OP didn't bite (yet). Will delete this, if the OP does give the details of how they eventually solved the problem. – Jyrki Lahtonen Nov 18 '13 at 5:53