# Finding Inverse of a Polynomial in a Galois Ring $GR(2^k,d)$

I'm trying to find the inverse of the polynomial ( 7X^2 + 1 ) in the Galois ring ( GR(2^3, 3) ) where the modulus polynomial is ( h(X) = X^3 + X + 1 ). However, I keep encountering the error "Flint exception (Impossible inverse): Cannot invert modulo (2 4)."

Here is the SageMath code I'm using:

```
k = 3
d = 3
R = IntegerModRing(2**k)
GR = PolynomialRing(R, 'X')
X = GR.gen()
h = X^3 + X + 1
GaloisRing = GR.quotient(h, 'X')
poly = GaloisRing(7*X^2 + 1)
poly_elem = poly.lift()
poly_modulus = h
g, u, v = xgcd(poly_elem, poly_modulus)
if g != 1: raise ValueError(f"The element {elem} does not have an inverse in the Galois ring.")
inverse = GaloisRing(u % poly_modulus)
```

Then I get an error:

```
Flint exception (Impossible inverse):Cannot invert modulo 2*4
------------------------------------------------------------------------
(no backtrace available)
------------------------------------------------------------------------
Unhandled SIGABRT: An abort() occurred.
This probably occurred because a *compiled* module has a bug
in it and is not properly wrapped with sig_on(), sig_off().
Python will now terminate.
```

**Questions:**
1. Why am I encountering this error when trying to find the inverse of ( 7X^2 + 1 ) in ( \text{GR}(2^3, 3) )?
2. Is there a mistake in how I'm defining the Galois ring or using the extended Euclidean algorithm?
3. How can I correctly find the inverse of \7X^2 + 1 ) in this Galois ring?