- #1
Joelly
- 3
- 0
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SUMMARY
The discussion focuses on the isomorphism between the algebraic structure of numbers of the form $a + b\sqrt{2}$ and the matrix representation $\begin{bmatrix} a & 2b \\ b & a \end{bmatrix}$. It establishes that this mapping is both one-to-one and onto. The preservation of operations under both multiplication and addition is demonstrated by comparing the results of these operations in both forms, confirming that the mapping is indeed an isomorphism.
PREREQUISITES- Understanding of algebraic structures, specifically fields and rings.
- Familiarity with matrix multiplication and properties of matrices.
- Knowledge of isomorphisms in abstract algebra.
- Basic proficiency in manipulating expressions involving square roots.
- Study the properties of isomorphisms in abstract algebra.
- Learn about matrix representations of algebraic structures.
- Explore the concept of fields, particularly quadratic fields like $\mathbb{Q}(\sqrt{2})$.
- Investigate applications of matrix algebra in solving linear equations.
Mathematicians, students of abstract algebra, and anyone interested in the relationship between algebraic numbers and matrix representations.
- #2
HOI
- 921
- 2
An obvious "guess", practically forced by the way the problem is stated, is the function that maps $a+ b\sqrt{2}$ to $\begin{bmatrix} a & 2b \\ b & a \end{bmatrix}$. Is that an isomorphism? That is, is it "one to one", is it "onto", and does it "preserve the operation".
The first two are simple. To prove that this "preserves the operation" you need to take two members of the first set, $a+ b\sqrt{2}$ and $x+ y\sqrt{2}$, which map to $\begin{bmatrix}a & 2b \\ b & a\end{bmatrix}$ and $\begin{bmatrix}x & 2y \\ y & a\end{bmatrix}$ and show that the product $(a+ b\sqrt{2})(x+ y\sqrt{2})$ maps to the product $\begin{bmatrix}a & 2b \\ b & a \end{bmatrix}\begin{bmatrix}x & 2y \\ y & x\end{bmatrix}$.
Do those two multiplications and compare them.
To determine whether they are also isomorphic under addition, add those pairs and compare them.
The first two are simple. To prove that this "preserves the operation" you need to take two members of the first set, $a+ b\sqrt{2}$ and $x+ y\sqrt{2}$, which map to $\begin{bmatrix}a & 2b \\ b & a\end{bmatrix}$ and $\begin{bmatrix}x & 2y \\ y & a\end{bmatrix}$ and show that the product $(a+ b\sqrt{2})(x+ y\sqrt{2})$ maps to the product $\begin{bmatrix}a & 2b \\ b & a \end{bmatrix}\begin{bmatrix}x & 2y \\ y & x\end{bmatrix}$.
Do those two multiplications and compare them.
To determine whether they are also isomorphic under addition, add those pairs and compare them.
- #3
Joelly
- 3
- 0
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