2020 IB Extended Essays

2.0 Quadratic Rings

2.1 Algebraic Integers and Quadratic Rings In mathematics, a ring is a set of elements that satisfies the properties of commutativity, associativity, identity, and inverse under addition, the property of associativity under multiplication, and the property of distributivity. Algebraic integers of the form + √ where , , ∈ ℤ and is non-square make up a quadratic ring ℤ�√ � (Weisstein, Eric W [unknown]). However, because elements other than zero do not have inverses, it is not a field. In this investigation of Pell’s equations and Fermat’s Sandwich Theorem, the basic definitions above are sufficient for our understanding of the problem. In Pell’s equations, the general ring ℤ�√ � and the example rings of ℤ�√ 7 � and ℤ�√ 13 � will be examined. In Fermat’s Sandwich Theorem, arithmetic in the quadratic ring ℤ�√− 2 � and the isomorphism between congruence ring ℤ / 8ℤ and ℤ will require the same basic understanding of the definition of rings.

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