We need to find the greatest common divisor (GCD) of the polynomials
$f(x) = x^5 + 1$ $g(x) = x^2 + 1$ in the field $\mathbb{F}_3$ (modulo 3) using the Extended Euclidean Algorithm.

Step 1: Reduce the Polynomials Modulo 3

Since we are working in $\mathbb{F}_3$, we reduce coefficients modulo 3. However, both polynomials already have coefficients in $\mathbb{F}_3$, so they remain:

$f(x) = x^5 + 1$ $g(x) = x^2 + 1$

Step 2: Perform Polynomial Division

We divide $f(x)$ by $g(x)$ using polynomial division in $\mathbb{F}_3$.

Long Division of $f(x)$ by $g(x)$

1. First term: Divide $x^5$ by $x^2$, which gives $x^3$.

2. Multiply: $x^3 \cdot (x^2 + 1) = x^5 + x^3$.

3. Subtract: $(x^5 + 1) - (x^5 + x^3) = -x^3 + 1 \equiv 2x^3 + 1 \pmod{3}.$

4. Next term: Divide $2x^3$ by $x^2$, which gives $2x$.

5. Multiply: $2x \cdot (x^2 + 1) = 2x^3 + 2x$.

6. Subtract: $(2x^3 + 1) - (2x^3 + 2x) = -2x + 1 \equiv x + 1 \pmod{3}.$

Now, the remainder is $x + 1$.

Step 3: Apply Euclidean Algorithm

We now compute $\gcd(x^2 + 1, x + 1)$ using the same method.

1. Divide $x^2 + 1$ by $x + 1$:

   • First term: $x^2 / x = x$.
   • Multiply: $x(x + 1) = x^2 + x$.
   • Subtract: $(x^2 + 1) - (x^2 + x) = -x + 1 \equiv x + 1 \pmod{3}.$
   • The remainder is $x + 1$.

2. Now compute $\gcd(x + 1, x + 1)$:

   • Since both are the same, we conclude: $\gcd(x^5 + 1, x^2 + 1) = x + 1.$

Final Answer

$\gcd(x^5 + 1, x^2 + 1) = x + 1 \quad \text{in } \mathbb{F}_3.$

---

Would you like a step-by-step breakdown of the polynomial division?

Related Questions

1. How does the Extended Euclidean Algorithm work for polynomials in general fields?
2. What is the difference between Euclidean Algorithm and Extended Euclidean Algorithm?
3. How do we find modular inverses in polynomial fields like $\mathbb{F}_3[x]$?
4. How can we compute GCDs of polynomials in larger fields like $\mathbb{F}_5$ or $\mathbb{F}_7$?
5. How does polynomial division work over finite fields?

Tip: The Euclidean algorithm for polynomials works similarly to numbers—keep dividing and subtracting until you reach the remainder 0!