# Circle Group

The real (unit) circle $C$ is defined algebraically by

$$
C = \{(x,y) \in \mathbb{R}^2 \mid x^2 + y^2 = 1\}.
$$

The definition only uses addition and multiplication, so it makes sense over any field.

For a field $\mathbb{F}$, we define the **$\mathbb{F}$-circle** as

$$
C(\mathbb{F}) = \{(x,y) \in \mathbb{F}^2 \mid x^2 + y^2 = 1\}.
$$

### Task

Using Sage, define $C(\mathbb{F}_p)$ for $p \equiv 3 \pmod{4}$

- Define $p$

p = 7

- Define $\mathbb{F}_p$

F = FiniteField(p)

- Define $C(\mathbb{F}_p)$

# Define empty set
C_Fp = []

# Loop over elements and add roots
for x in F:
    for y in F:
        if x^2 + y^2 == 1:
            C_Fp.append((x, y))