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Unique Factorization Domain Problems and Solutions


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  • Every Group of Order 20449 is an Abelian GroupEvery Group of Order 20449 is an Abelian Group Prove that every group of order $20449$ is an abelian group.   Outline of the Proof Note that $20449=11^2 \cdot 13^2$. Let $G$ be a group of order $20449$. We prove by Sylow's theorem that there are a unique Sylow $11$-subgroup and a unique Sylow $13$-subgroup of […]
  • Extension Degree of Maximal Real Subfield of Cyclotomic FieldExtension Degree of Maximal Real Subfield of Cyclotomic Field Let $n$ be an integer greater than $2$ and let $\zeta=e^{2\pi i/n}$ be a primitive $n$-th root of unity. Determine the degree of the extension of $\Q(\zeta)$ over $\Q(\zeta+\zeta^{-1})$. The subfield $\Q(\zeta+\zeta^{-1})$ is called maximal real subfield.   Proof. […]
  • Dimension of Null Spaces of Similar Matrices are the SameDimension of Null Spaces of Similar Matrices are the Same Suppose that $n\times n$ matrices $A$ and $B$ are similar. Then show that the nullity of $A$ is equal to the nullity of $B$. In other words, the dimension of the null space (kernel) $\calN(A)$ of $A$ is the same as the dimension of the null space $\calN(B)$ of […]
  • Two Quadratic Fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are Not IsomorphicTwo Quadratic Fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are Not Isomorphic Prove that the quadratic fields $\Q(\sqrt{2})$ and $\Q(\sqrt{3})$ are not isomorphic.   Hint. Note that any homomorphism between fields over $\Q$ fixes $\Q$ pointwise. Proof. Assume that there is an isomorphism $\phi:\Q(\sqrt{2}) \to \Q(\sqrt{3})$. Let […]
  • Prove the Cauchy-Schwarz InequalityProve the Cauchy-Schwarz Inequality Let $\mathbf{a}, \mathbf{b}$ be vectors in $\R^n$. Prove the Cauchy-Schwarz inequality: \[|\mathbf{a}\cdot \mathbf{b}|\leq \|\mathbf{a}\|\,\|\mathbf{b}\|.\]   We give two proofs. Proof 1 Let $x$ be a variable and consider the length of the vector […]
  • Find the Inverse Matrix of a Matrix With FractionsFind the Inverse Matrix of a Matrix With Fractions Find the inverse matrix of the matrix \[A=\begin{bmatrix} \frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\[6 pt] \frac{6}{7} &\frac{2}{7} &-\frac{3}{7} \\[6pt] -\frac{3}{7} & \frac{6}{7} & -\frac{2}{7} \end{bmatrix}.\]   Hint. You may use the augmented matrix […]
  • The Polynomial Rings $\Z[x]$ and $\Q[x]$ are Not IsomorphicThe Polynomial Rings $\Z[x]$ and $\Q[x]$ are Not Isomorphic Prove that the rings $\Z[x]$ and $\Q[x]$ are not isomoprhic.   Proof. We give three proofs. The first two proofs use only the properties of ring homomorphism. The third proof resort to the units of rings. If you are familiar with units of $\Z[x]$, then the […]
  • Group of Invertible Matrices Over a Finite Field and its StabilizerGroup of Invertible Matrices Over a Finite Field and its Stabilizer Let $\F_p$ be the finite field of $p$ elements, where $p$ is a prime number. Let $G_n=\GL_n(\F_p)$ be the group of $n\times n$ invertible matrices with entries in the field $\F_p$. As usual in linear algebra, we may regard the elements of $G_n$ as linear transformations on $\F_p^n$, […]

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