 Show that an integer prime p remains prime in Z[
√
−2] if p is congruent
to 5 or 7 mod 8

To show that an integer prime \( p \) remains prime in \( \mathbb{Z}[\sqrt{-2}] \) if \( p \equiv 5 \) or \( 7 \mod 8 \), we will use the properties of norms in this ring.

### Step 1: Norms in \( \mathbb{Z}[\sqrt{-2}] \)

In \( \mathbb{Z}[\sqrt{-2}] \), the norm \( N(a + b\sqrt{-2}) \) of an element \( a + b\sqrt{-2} \) is defined as:
\[
N(a + b\sqrt{-2}) = a^2 + 2b^2.
\]
This norm maps elements from \( \mathbb{Z}[\sqrt{-2}] \) to non-negative integers.

### Step 2: Primality Conditions

For \( p \) to remain prime in \( \mathbb{Z}[\sqrt{-2}] \), it must not be expressible as a product of non-unit elements in this ring. If \( p \) can be factored in \( \mathbb{Z}[\sqrt{-2}] \), then there exist \( x, y \in \mathbb{Z}[\sqrt{-2}] \) such that:
\[
p = xy \quad \text{with } N(x), N(y) < N(p) = p^2.
\]

### Step 3: Checking Congruences

#### Case 1: \( p \equiv 5 \mod 8 \)

Suppose \( p = 8k + 5 \) for some integer \( k \). We can compute the quadratic residues modulo \( p \):

The quadratic residues modulo \( p \) can be derived using \( a^2 + 2b^2 \equiv 0 \mod p \). 

Consider \( a^2 + 2b^2 \equiv 0 \mod p \). This can be rearranged to find \( a \) and \( b \).

Now note that since \( 2 \) is not a quadratic residue modulo \( p \) when \( p \equiv 5 \mod 8 \), the equation \( x^2 \equiv -2 \mod p \) has no solutions, meaning \( p \) cannot be factored into the form \( (a + b\sqrt{-2})(c + d\sqrt{-2}) \).

#### Case 2: \( p \equiv 7 \mod 8 \)

For \( p = 8k + 7 \), the analysis is similar:

The equation \( a^2 + 2b^2 \equiv 0 \mod p \) again leads us to conclude that \( -2 \) is also not a quadratic residue modulo \( p \). Therefore, the quadratic residues will not allow \( p \) to be factored as before.

### Conclusion

Since \( p \equiv 5 \mod 8 \) and \( p \equiv 7 \mod 8 \) lead to the conclusion that \( p \) cannot be expressed as a product of two non-unit elements in \( \mathbb{Z}[\sqrt{-2}] \), we conclude that:

If \( p \equiv 5 \) or \( 7 \mod 8 \), then \( p \) remains prime in \( \mathbb{Z}[\sqrt{-2}] \).

### Questions for Further Exploration

1. What is the significance of quadratic residues in number theory?
2. Can you provide examples of primes that are congruent to \( 5 \mod 8 \) and \( 7 \mod 8 \)?
3. How does the ring \( \mathbb{Z}[\sqrt{-d}] \) differ for various values of \( d \)?
4. What are the implications of a prime remaining prime in a number field?
5. How do norms help in understanding the structure of integers in quadratic fields?

### Tip

Always check the properties of quadratic residues when dealing with primes in rings involving square roots.

Show that an integer prime p remains prime in Z[√−2] if p is congruent to 5 or 7 mod 8.

Explore how integer primes p remain prime in the ring Z[√−2] if p is congruent to 5 or 7 modulo 8. This solution utilizes properties of norms and quadratic residues to establish the primality conditions in quadratic fields. Ideal for undergraduate students studying number theory.

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