In 2023, Ray and Weston defined the notion of diophantine stability at ℓ for an elliptic curve E/K defined over a number field K and a prime ℓ. We provide a brief discussion of Galois theory and algebraic number theory, while building up intuition and implications for possible answers to the following question: For a fixed elliptic curve E/K and a fixed prime ℓ > 3, what number fields K allow E/K to satisfy the property of being diophantine stable at ℓ?
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