## Example of an Infinite Algebraic Extension

## Problem 499

Find an example of an infinite algebraic extension over the field of rational numbers $\Q$ other than the algebraic closure $\bar{\Q}$ of $\Q$ in $\C$.

Add to solve laterFind an example of an infinite algebraic extension over the field of rational numbers $\Q$ other than the algebraic closure $\bar{\Q}$ of $\Q$ in $\C$.

Add to solve laterLet $T:\R^2 \to \R^2$ be a linear transformation of the $2$-dimensional vector space $\R^2$ (the $x$-$y$-plane) to itself which is the reflection across a line $y=mx$ for some $m\in \R$.

Then find the matrix representation of the linear transformation $T$ with respect to the standard basis $B=\{\mathbf{e}_1, \mathbf{e}_2\}$ of $\R^2$, where

\[\mathbf{e}_1=\begin{bmatrix}

1 \\

0

\end{bmatrix}, \mathbf{e}_2=\begin{bmatrix}

0 \\

1

\end{bmatrix}.\]

Let $G$ be an abelian group.

Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively.

Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$.

Also determine whether the statement is true if $G$ is a non-abelian group.

Add to solve laterProve that if $2^n-1$ is a Mersenne prime number, then

\[N=2^{n-1}(2^n-1)\]
is a perfect number.

On the other hand, prove that every even perfect number $N$ can be written as $N=2^{n-1}(2^n-1)$ for some Mersenne prime number $2^n-1$.

Add to solve laterProve that every finite group having more than two elements has a nontrivial automorphism.

(*Michigan State University, Abstract Algebra Qualifying Exam*)

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Let $G$ be a finite group and let $A, B$ be subsets of $G$ satisfying

\[|A|+|B| > |G|.\]
Here $|X|$ denotes the cardinality (the number of elements) of the set $X$.

Then prove that $G=AB$, where

\[AB=\{ab \mid a\in A, b\in B\}.\]

Let

\[D=\begin{bmatrix}

d_1 & 0 & \dots & 0 \\

0 &d_2 & \dots & 0 \\

\vdots & & \ddots & \vdots \\

0 & 0 & \dots & d_n

\end{bmatrix}\]
be a diagonal matrix with distinct diagonal entries: $d_i\neq d_j$ if $i\neq j$.

Let $A=(a_{ij})$ be an $n\times n$ matrix such that $A$ commutes with $D$, that is,

\[AD=DA.\]
Then prove that $A$ is a diagonal matrix.

Let $\zeta_8$ be a primitive $8$-th root of unity.

Prove that the cyclotomic field $\Q(\zeta_8)$ of the $8$-th root of unity is the field $\Q(i, \sqrt{2})$.

Let $G, H, K$ be groups. Let $f:G\to K$ be a group homomorphism and let $\pi:G\to H$ be a surjective group homomorphism such that the kernel of $\pi$ is included in the kernel of $f$: $\ker(\pi) \subset \ker(f)$.

Define a map $\bar{f}:H\to K$ as follows.

For each $h\in H$, there exists $g\in G$ such that $\pi(g)=h$ since $\pi:G\to H$ is surjective.

Define $\bar{f}:H\to K$ by $\bar{f}(h)=f(g)$.

**(a)** Prove that the map $\bar{f}:H\to K$ is well-defined.

**(b)** Prove that $\bar{f}:H\to K$ is a group homomorphism.

Suppose that $\alpha$ is a rational root of a monic polynomial $f(x)$ in $\Z[x]$.

Prove that $\alpha$ is an integer.

Let $G$ be a finite group and let $S$ be a non-empty set.

Suppose that $G$ acts on $S$ freely and transitively.

Prove that $|G|=|S|$. That is, the number of elements in $G$ and $S$ are the same.

Let

\[P=(2, \sqrt{10})=\{a+b\sqrt{10} \mid a, b \in \Z, 2|a\}\]
be an ideal of the ring

\[\Z[\sqrt{10}]=\{a+b\sqrt{10} \mid a, b \in \Z\}.\]
Then determine the quotient ring $\Z[\sqrt{10}]/P$.

Is $P$ a prime ideal? Is $P$ a maximal ideal?

Determine whether there exists a nonsingular matrix $A$ if

\[A^4=ABA^2+2A^3,\]
where $B$ is the following matrix.

\[B=\begin{bmatrix}

-1 & 1 & -1 \\

0 &-1 &0 \\

2 & 1 & -4

\end{bmatrix}.\]

If such a nonsingular matrix $A$ exists, find the inverse matrix $A^{-1}$.

(*The Ohio State University, Linear Algebra Final Exam Problem*)

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Let

\[A=\begin{bmatrix}

1 & -14 & 4 \\

-1 &6 &-2 \\

-2 & 24 & -7

\end{bmatrix} \quad \text{ and }\quad \mathbf{v}=\begin{bmatrix}

4 \\

-1 \\

-7

\end{bmatrix}.\]
Find $A^{10}\mathbf{v}$.

You may use the following information without proving it.

The eigenvalues of $A$ are $-1, 0, 1$. The eigenspaces are given by

\[E_{-1}=\Span\left\{\, \begin{bmatrix}

3 \\

-1 \\

-5

\end{bmatrix} \,\right\}, \quad E_{0}=\Span\left\{\, \begin{bmatrix}

-2 \\

1 \\

4

\end{bmatrix} \,\right\}, \quad E_{1}=\Span\left\{\, \begin{bmatrix}

-4 \\

2 \\

7

\end{bmatrix} \,\right\}.\]

(*The Ohio State University, Linear Algebra Final Exam Problem*)

Let $A$ be a square matrix and its characteristic polynomial is given by

\[p(t)=(t-1)^3(t-2)^2(t-3)^4(t-4).\]
Find the rank of $A$.

(*The Ohio State University, Linear Algebra Final Exam Problem*)

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Diagonalize the matrix

\[A=\begin{bmatrix}

1 & 1 & 1 \\

1 &1 &1 \\

1 & 1 & 1

\end{bmatrix}.\]
Namely, find a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.

(*The Ohio State University, Linear Algebra Final Exam Problem*)

For which values of constants $a, b$ and $c$ is the matrix

\[A=\begin{bmatrix}

7 & a & b \\

0 &2 &c \\

0 & 0 & 3

\end{bmatrix}\]
diagonalizable?

(*The Ohio State University, Linear Algebra Final Exam Problem*)

Let $P_2$ be the vector space of all polynomials with real coefficients of degree $2$ or less.

Let $S=\{p_1(x), p_2(x), p_3(x), p_4(x)\}$, where

\begin{align*}

p_1(x)&=-1+x+2x^2, \quad p_2(x)=x+3x^2\\

p_3(x)&=1+2x+8x^2, \quad p_4(x)=1+x+x^2.

\end{align*}

**(a)** Find a basis of $P_2$ among the vectors of $S$. (Explain why it is a basis of $P_2$.)

**(b)** Let $B’$ be the basis you obtained in part (a).

For each vector of $S$ which is not in $B’$, find the coordinate vector of it with respect to the basis $B’$.

(*The Ohio State University, Linear Algebra Final Exam Problem*)

**(a)** Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}

x \\

y \\

z \\

w

\end{bmatrix}$ satisfying

\[2x+4y+3z+7w+1=0.\]
Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

**(b)** Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}

x \\

y \\

z \\

w

\end{bmatrix}$ satisfying

\[2x+4y+3z+7w=0.\]
Determine whether $S$ is a subspace of $\R^4$. If so prove it. If not, explain why it is not a subspace.

(These two problems look similar but note that the equations are different.)

(*The Ohio State University, Linear Algebra Final Exam Problem*)

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