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Corollary

If

1. \( \forall s \in S, s\in A_1 \cup A_2 \cup \cdots \cup A_n \, \land \, A_1,A_2,\ldots,A_n \) are pairwise disjoint
2. \( s\in A_i \implies s \in B_i \)
3. \( \forall s \in S, s\in B_1 \cup B_2 \cup \cdots \cup B_n \, \land \, B_1,B_2,\ldots,B_n \) are pairwise disjoint

Then \( \forall s \in S, s\in B_i \implies s \in A_i \)

Proposition Suppose there exists a nontrivial element of \( \mathbb { Z } [ \sqrt { d } ] ^ { \times , 1 } \). Then every element of \( \mathbb { Z } [ \sqrt { d } ] ^ { \times , 1 } \) is of the form \( \pm \epsilon ^ { n } \) for some \( n \) in \( \mathbb { Z } \), where \( \epsilon \) is the fundamental 1-unit.

Proposition If \( \alpha \) is an algebraic integer of degree two, then \( \mathbb Z [\alpha] \) is equal to the set of complex numbers of the form \( x + y\alpha \), where x and y are integers.

Theorem Suppose that unique factorization holds in \( \mathbb { Z } [ \alpha ] \), and let p be an integer prime such that the polynomial \( P ( x , 1 ) = x ^ { 2 } - b x + c \) has a root mod p. Then there exist integers x and y such that \( P (x, y) = p \). Conversely, if there exist such x and y, then \( x^2 - bx + c \) has a root mod p.

Proposition (Prime in \( {\mathbb Z [i] } \)) Primes in \( {\mathbb Z [i] } \) are either of the form a +bi, where \( a^2 +b^2 \) is an integer prime, or, q and its associate, where q is an integer prime that is not the sum of two squares.

Proposition The ring \( \mathbb Z[i] \) is a Euclidean domain.

Fermat’s Two-Square Theorem Every prime congruent to \( 1 \bmod 4 \) is the sum of two squares.

Proposition a is a primitive root modulo n then it is also a primitive root modulo d, for any d dividing n.