2020 IB Extended Essays

Therefore, in the imaginary quadratic ring ℤ�√− 2 � , 3 = 2 + 2 is uniquely factorized into 3 = � + √ 2 �� − √ 2 � . can be expressed as a unique decomposition of irreducible factors 1 , 2 , 3 , ⋯ : = 1 1 2 2 3 3 ⋯ = ∏ ∈ ⇔ 3 = ∏ 3 ∈ 4.2 Quadratic Residues and Relative Primality of + √ 2 and − √ 2 If + √ 2 and − √ 2 are mutual primes (numbers that do not share any other factors than 1 in ℤ�√− 2 � , the equation 3 = 2 + 2 = ( + √ 2 )( − √ 2 ) could be simplified further. Therefore, the GCD (greatest common divisor) of the two algebraic integers needs to be examined. � + √ 2 , − √ 2 � = � + √ 2 , � + √ 2 � − � − √ 2 �� , ( ∵ ( , ) = ( , + )) Therefore, � + √ 2 , − √ 2 ��√ 2 , so that if √ 2 | , then + √ 2 and − √ 2 are relatively prime. However, because is an integer, there are two possible cases depending on the parity of : � + √ 2 , − √ 2 � = √ 2 if and only if is even, and � + √ 2 , − √ 2 � = 1 otherwise. Now, the parity of and need to be determined, via quadratic residues, reduction ad absurdum, and proof by contradiction. is called a quadratic residue modulo if ∃ , . . 2 ≡ ( ) . The reduction ad absurdum explains that the modulo relationship that holds in congruence ring ℤ / 8ℤ also holds in ℤ , since ℤ / 8ℤ and ℤ are isomorphic (Voisin, J 2016). For proof by contradiction, the is assumed to be even; If is even (i.e. = 2 , ∈ ℤ ), � + √ 2 , − √ 2 � = ( + √ 2 , 2 √ 2 ) � + √ 2 , − √ 2 � = ( + √ 2 , √ 2 3 )

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