Chapter 11, Polynomials and Finite Fields Video Solutions, Foundations of coding: theory and applications of error-correcting codes

Problem 1

Let $F$ be a field of characteristic $p$. For each natural number $n$, denote
$$
n=1+1+\cdots+1 \quad \text { ( } n \text { summands). }
$$

In particular, in $F$, we have $p=0, p+1=1$, etc.
Verify that for all $n=0,1, \ldots, p(=0)$, we can define binomial coeficients in $F$ in the usual manner:
$$
\left(\begin{array}{l}
n \\
k
\end{array}\right)=\frac{n(n-1) \cdots(n-k+1)}{k(k-1) \cdots 1}=n(n-1) \cdots(n-k+1)[k(k-1) \cdots 1]^{-1}
$$
if $0<k<p$, and
$$
\left(\begin{array}{l}
n \\
0
\end{array}\right)=1, \quad\left(\begin{array}{l}
p \\
p
\end{array}\right)=1 .
$$

Verify that the following well-known formula is valid:
$$
\left(\begin{array}{l}
n \\
k
\end{array}\right)+\left(\begin{array}{c}
n \\
k-1
\end{array}\right)=\left(\begin{array}{c}
n+1 \\
k
\end{array}\right) .
$$

Oswaldo Jiménez

Numerade Educator

07:17

Problem 2

Prove that a field of characteristic $p$ has the following properties.
(1) The usual binomial theorem:
$$
(a+b)^n=\sum_{k=0}^n\left(\begin{array}{l}
n \\
k
\end{array}\right) a^k b^{n-k}
$$
holds for all $n=1,2, \ldots, p$ and arbitrary elements $a, b$; beware that the exponents are interpreted as ordinary natural numbers, not as field elements: $a^i=a \times a \times \cdots \times d$ ( $i$ factors). [Hint: use mathematical induction on $n$, applying the formula in 11A.]
(2) For each polynomial $f(x)$ over $Z_p$ we have
$$
f\left(x^p\right)=[f(x)]^p .
$$

Joanna Quigley

Numerade Educator

17:20

Problem 3

The formal derivative of a polynomial $a(x)=a_0+a_1 x+a_2 x^2+$ $\cdots+a_n x^n$ is defined by $a^{\prime}(x)=a_1+2 a_2 x+3 a_3 x^2+\cdots+n a_n x^{n-1}$ (where $2=1+1$, etc.). Prove that the "usual" formulas for derivatives are true:
(1) $[a(x)+b(x)]^{\prime}=a^{\prime}(x)+b^{\prime}(x)$,
(2) $[a(x) b(x)]^{\prime}=a^{\prime}(x) b(x)+a(x) b^{\prime}(x)$.

Leon Druch

Numerade Educator

02:28

Problem 4

Multiple zeros. An element $a$ is called a multiple zero of a polynomial $f(x)$ provided that $(x-a)^k$ divides $f(x)$ for some $k \geq 2$.
(1) Prove that each multiple zero of $f(x)$ is a zero of the formal derivative $f^{\prime}(x)$.
(2) Find a polynomial over $Z_2$ with no multiple zeros whose derivative has a zero.
(3) Prove that every common zero of $f(x)$ and $f^{\prime}(x)$ is a multiple zero of $f(x)$.

Nick Johnson

Numerade Educator

Problem 5

Prove that in a field of characteristic 2:
(1) Every element has a square root (i.e., for each $a$, there exists $b$ with $a=b^2$ ). [Hint: if $c$ is primitive in $G F\left(2^m\right)$, then $c^{2-}=c$; thus, $c$ has the square root $c^{2--1}$, and then each $c^i$ has a square root too.]
(2) The formal derivative of every polynomial is a perfect square (i.e., for each polynomial $f(x)$, there exists a polynomial $g(x)$ with $f^{\prime}(x)=$ $[g(x)]^2$ ). [Hint: since $\left(x^n\right)^{\prime}=n x^{n-1}=0$ for $n$ even, the formal derivative has only even powers. Using (1), write $f^{\prime}(x)=f_0^2+f_2^2 x^2+$ $f_4^2 x^4+\cdots$ and then put $g(x)=f_0+f_2 x+f_4 x^2+\cdots$ ]

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Problem 6

Verify that orders of elements of $G F\left(2^m\right)$ are always odd numbers. More in general: if a field of $q$ elements has the property that some of its extensions contain an element of order $n$, verify that $q$ and $n$ are relatively prime. Compare with Corollary 11.7.

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00:44

Problem 7

Formal power series. Just as (finite) words can be expressed by polynomials, infinite words correspond to a so-called formal power series. In other words, given a field $F$, a formal power series over $F$ in one indeterminate $x$ is an expression of the form $\sum_{i=0}^{\infty} a_i x^i$, where $a_0, a_1, a_2, \ldots$ are elements of $F$. The summands $a_i x^i$ with $a_i=0$ can be omitted from the expression. (Thus, in particular, every polynomial is a formal power series.) The question of convergence, discussed so much in the case $F=\mathbf{R}$, never arises in finite fields; thus, a formal power series is not a function.
(1) Verify that all formal power series form a ring under the usual addition
$$
\sum_{i=0}^{\infty} a_i x^i+\sum_{i=0}^{\infty} b_i x^i=\sum_{i=0}^{\infty}\left(a_i+b_i\right) x^i
$$
and multiplication
$$
\left(\sum_{i=0}^{\infty} a_i x^i\right)\left(\sum_{i=0}^{\infty} b_i x^i\right)=\sum_{i=0}^{\infty}\left(a_i b_0+a_{i-1} b_1+\cdots+a_0 b_i\right) x^i .
$$
(2) A remarkable fact: the well-known formula for the sum of a geometric series is valid even in the formal setting: for each element $a$,
$$
\sum_{i=0}^{\infty} a^i x^i=\frac{1}{1-a x} .
$$
More precisely, the product of the power series $\sum a^i x^i$ and $1-a x$ is 1 . Prove it. [Hint: multiply $\sum a^i x^i(1-a x)=\sum a^i x^i-\sum a^{i+1} x^{i+1}$.]

Monica Miller

Numerade Educator

Problem 8

Prove that polynomials over any field have the following properties:
(1) The following formula
$$
x^{k m}-1=\left(x^m-1\right) \sum_{i=0}^{k-1}\left(x^m\right)^i
$$
holds. Therefore,
$$
m \text { divides } n \quad \Longrightarrow \quad x^m-1 \text { divides } x^n-1 \text {. }
$$
(2) Conversely, whenever $x^m-1$ divides $x^n-1$, prove that $m$ divides $n$.

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Problem 9

Let $f(x)$ be an irreducible polynomial over a field $F$. Prove that $f(x)$ is the minimal polynomial (with respect to $F$ ) of any zero $f(x)$ has in any extension of $F$. Use this fact to describe the minimal polynomials of all elements of $G F(16)$ w.r.t. $G F(4)$. Can you apply Theorem 11.6?

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09:17

Problem 10

Prove the following:
(1) $x^8-1$ factors as the product of all binary irreducible polynomials whose degree divides 3 (see Figure 4 of 11.6).
(2) More in general, the product of all binary irreducible polynomials whose degree divides $n$ is equal to $x^{2^2}-1$.
(3) Generalize (2) from $Z_2$ to any finite field.

P Krishnamurthy

Numerade Educator

Problem 11

Find a primitive element of $Z_3[x] / x^2+1$.

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Problem 12

What is an algebraic extension $F[x] / \bmod p(x)$ in the case $p(x)$ has degree 1 ?

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Problem 13

Prove that in a field of characteristic $p$, no element has an order divisible by $p$.

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Problem 14

Prove that in a field of characteristic $p$, every polynomial $f(x)$ fulfils
$$
[f(x)]^{p^{m-}}=f_0^{p^m}+f_1^{p^m} x^{p^{m+}}+\cdots+f_n^{p^{m+}} x^{n p^m} .
$$
Conclude that for polynomials $f(x)$ over $\mathrm{Z}_p$,
$$
[f(x)]^{p^{m=}}=f\left(x^{p^m}\right) .
$$

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01:58

Problem 15

Find an extension of $\mathrm{Z}_2$ in which $x^9-1$ factors into linear factors.

Ankit Gupta

Numerade Educator

Problem 16

Prove that if $G F\left(2^k\right)$ extends $G F\left(2^n\right)$, then $k$ is divisible by $n$. Find all fields which are extended by $G F(16)$.

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04:34

Problem 17

Show that the polynomial $x^2-1$ has more than two zeros in $\mathrm{Z}_{15}$. Compare this with Remark (1) in 11.1.

Gabriel Eduok

Numerade Educator

Problem 18

$\mathrm{What}$ are the orders of all elements of $\operatorname{GF}(27)$ and $G F(32)$ ?

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Problem 19

An irreducible polynomial $p(x)$ over a field $F$ is called primitive if the indeterminate $\alpha$ is a primitive element of $F[x] / \bmod p(x)$.
(1) Find a nonprimitive irreducible polynomial of degree 2 over $Z_3$, and find a primitive polynomial of degree 2 over $Z_3$.
(2) Prove that every finite field has primitive polynomials of all degrees.

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Problem 20

Construct $G F(16)$ as an algebraic extension of $G F(4)$.

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