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Proposition If \( I + J = \langle 1 \rangle \), then \( I \cap J = I J \)

Third Isomorphism Theorem for Groups Let \( G \) be a group and let \( H \) and \( K \) be normal subgroups of \( G \), with \( H \leq K \). Then

1. \( K / H \unlhd G / H \)
2. \( ( G / H ) / ( K / H ) \cong G / K \)

Second Isomorphism Theorem for Groups Let \( G \) be a group, \( H \leq G \) and \( N \unlhd G \). Then

1. \( HN \leq G \)
2. \( H \cap N \unlhd H \)
3. \( HN / N \cong H / ( H \cap N ) \)

First Isomorphism Theorem for Groups If \( \phi : G \rightarrow H \) is a homomorphism then

\[ G / \operatorname { ker } ( \phi ) \cong \operatorname { im } ( \phi ) \]

Proposition Suppose that \( \alpha \) has an eventually periodic continued fraction expansion. Then \( \alpha \) is a quadratic irrational.

Liouville’s theorem on diophantine approximation Let \( \alpha \) be an irrational number that is algebraic of degree \( d \). Then for any real number \( e > d \), there are at most finitely many rational numbers \( \frac { { p } } { q } \) such that \( \left| \frac { p } { q } - \alpha \right| < \frac { 1 } { q ^ { e } } \).

Theorem 3 For any squarefree d, there is a nontrivial integer solution to \( x^2 - d y^2=1 \)

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