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{\hfill{Fall 2021} \smallskip %
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{\LARGE Introduction to Algebraic Structures\\} 

{\LARGE Problem Set 7\\}
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{\LARGE Finite fields.}
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\noindent 1. Find all irreducible polynomials \mbox{$f(x)\in\Z_2[x]$} of degree 4 or lower. Use of computer (algebra) is encouraged.

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\noindent 2. Find the multiplication table of the quotient ring $\Z_2[x]/( x^3 + 1)$. Is it a field? 

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\noindent 3. Prove that the ring $\Z_5[x]/( x^2 + x + 1)$ is the field $\mathbb{F}_{25}$. Find a primitive element of $\mathbb{F}_{25}$ and all the powers of that element as (equivalence classes of) polynomials (as in the table on page 92 of the lecture notes).

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\noindent 4. Calculate in $\F_{27} = \Z_5[x]/( x^2 + x + 1)$:
$$\left( [x^4+2x^2+3]+[x-1]^{2021}+[x^2+2]\right)\cdot \left( [x^{19}-x]+[3x^3+x+2]\right).$$

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\noindent 5. Find \emph{all} the primitive elements of the field $\F_{16} = \Z_2[x]/( x^4 + x^3 + x^2 + x + 1)$. 
\\\hfill\vskip-30pt\begin{turn}{180}{\tiny{\it Hint:} How do you get all generators of a cyclic group if you know one of them? A certain Corollary 7.13 might help...}\hfill\end{turn}

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\noindent 6. Let $F$ be a finite field. Find $\sum\limits_{x\in F^*}x$.

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\noindent 7. Prove that in any finite field the product of two non-squares is a square. 

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\noindent 8. Prove that if the multiplicative group of a field is cyclic, then the field is finite.

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\noindent 9${^*}$. Let $F$ be a finite field, $n\in\N$, $k_1,\ldots,k_n,k\in F$. Which equations of the following form have a solution: $$\sum\limits_{i=1}^n k_i (x_i)^2 = k?$$

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\noindent 10${^{**}}$. Take a non-square $n\in\N$, an odd prime $p$, $R_n=\{a+\sqrt{n}\cdot b\mid a,b\in\Z\}$, and $I_p=\{a+\sqrt{n}\cdot b\in R_n\mid\, p\mid a\wedge p\mid b\}.$ Show that $I_p$ is an ideal of $R_n$ and find a necessary and sufficient condition in terms of $n$ and $p$ for $R_n/I_p$ to be a field. Is this field finite, and if so, how many elements does it have? 

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