Galois Field $2^3$ Multiplication of $111$ and $111$

Question

In Galois Field Multiplication is performed as a binary multiplication between the two binary values representing polynomials and then is reduced by a reduction polynomial

So if i have the binary value $111$ represented as $X^2 + X + 1$ being multiplied by itself, the textbook claims in $GF(2^3)$ the answer should be $X + 1$ if reduced by a polynomial $X^3 + X + 1$ or in binary $1011$

Here is my answer

$111 \times 111$

---

----$110001$ $\longrightarrow$ Value after binary multiplication

XOR $1011$ $\longrightarrow$ Reduction polynomial

---

at the last step after reduction I'm left with

-----$1011$

XOR $1011$

---

-----0000<----- value is zero then why does the table say $111 * 111$ in $GF(2^3)$ is $011$ or $X+1$

Answer 1

111x111 is not exactly the same in normal number multiplication

in this group, 111x111=10101

there is no carry, each digit is represented as the coefficient in polynomial, and it is computed modular 2

then 10101 mod 1011 = 11