# field-theory-2

by Yu · Published · Updated

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- A Linear Transformation from Vector Space over Rational Numbers to itself Let $\Q$ denote the set of rational numbers (i.e., fractions of integers). Let $V$ denote the set of the form $x+y \sqrt{2}$ where $x,y \in \Q$. You may take for granted that the set $V$ is a vector space over the field $\Q$. (a) Show that $B=\{1, \sqrt{2}\}$ is a basis for the […]
- If a Symmetric Matrix is in Reduced Row Echelon Form, then Is it Diagonal? Recall that a matrix $A$ is symmetric if $A^\trans = A$, where $A^\trans$ is the transpose of $A$. Is it true that if $A$ is a symmetric matrix and in reduced row echelon form, then $A$ is diagonal? If so, prove it. Otherwise, provide a counterexample. Proof. […]
- If $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent Matrix Let $A$ be a square matrix such that \[A^{\trans}A=A,\] where $A^{\trans}$ is the transpose matrix of $A$. Prove that $A$ is idempotent, that is, $A^2=A$. Also, prove that $A$ is a symmetric matrix. Hint. Recall the basic properties of transpose […]
- The Ideal Generated by a Non-Unit Irreducible Element in a PID is Maximal Let $R$ be a principal ideal domain (PID). Let $a\in R$ be a non-unit irreducible element. Then show that the ideal $(a)$ generated by the element $a$ is a maximal ideal. Proof. Suppose that we have an ideal $I$ of $R$ such that \[(a) \subset I \subset […]
- Determine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable Matrix Let $A$ be an $n\times n$ matrix with the characteristic polynomial \[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\] Assume that the matrix $A$ is diagonalizable. (a) Find the size of the matrix $A$. (b) Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue […]
- The Polynomial $x^p-2$ is Irreducible Over the Cyclotomic Field of $p$-th Root of Unity Prove that the polynomial $x^p-2$ for a prime number $p$ is irreducible over the field $\Q(\zeta_p)$, where $\zeta_p$ is a primitive $p$th root of unity. Hint. Consider the field extension $\Q(\sqrt[p]{2}, \zeta)$, where $\zeta$ is a primitive $p$-th root of […]
- If Eigenvalues of a Matrix $A$ are Less than $1$, then Determinant of $I-A$ is Positive Let $A$ be an $n \times n$ matrix. Suppose that all the eigenvalues $\lambda$ of $A$ are real and satisfy $\lambda <1$. Then show that the determinant \[ \det(I-A) >0,\] where $I$ is the $n \times n$ identity matrix. We give two solutions. Solution 1. Let […]
- Example of an Element in the Product of Ideals that Cannot be Written as the Product of Two Elements Let $I=(x, 2)$ and $J=(x, 3)$ be ideal in the ring $\Z[x]$. (a) Prove that $IJ=(x, 6)$. (b) Prove that the element $x\in IJ$ cannot be written as $x=f(x)g(x)$, where $f(x)\in I$ and $g(x)\in J$. Hint. If $I=(a_1,\dots, a_m)$ and $J=(b_1, \dots, b_n)$ are […]