%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\documentclass[12pt,a4paper,oneside]{amsart}
\usepackage[a4paper]{geometry}
\geometry{hmargin=1.8cm,top=2cm,bottom=2.5cm}

\usepackage{mathptmx} % Times font
%\DeclareMathAlphabet{\mathcal}{OMS}{cmsy}{m}{n} % but do not change mathcal symbols

\usepackage{enumerate}

\setlength{\parindent}{0pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\title{
Exam for ``Algebraic Structures'' (6ALGS) 
\\ 
summer semester 2023/2024
}

\maketitle

\thispagestyle{empty}

\section*{Questions set no. 6}

\vskip 20pt

\textbf{1.} 

Give definition of a prime field. Illustrate this notion by examples.

\vskip 20pt 

\textbf{2.}

Is it true that if $S$ is a subring of a ring $R$, then the additive group of 
$S$ is a subgroup of an additive group of $R$?

\vskip 20pt

\textbf{3.}

Let $\lambda \in \mathbb R$, and $M_\lambda$ is the set of all nonzero matrices
of the form
$$
\left(
\begin{matrix}
x         & y  \\
\lambda y & x
\end{matrix}
\right)
$$
where $x,y \in \mathbb R$. For which values of $\lambda$, $M_\lambda$ is:

a) a group

b) a simple group

with respect to the usual matrix multiplication?



\end{document}
