Math Problems Vault

Problem III.5.2.1

Prove that for any $d \mid n, d<n$ , the relation $X^{n}-1=\left(X^{d}-1\right) \Phi_{n}(X) h_{d}(X)$ holds, where $h_{d} \in \mathbb {Z}{}[X]$ .

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Problem III.5.2.2

Let $q$ be a positive integer $>1$ . By (10), $\Phi_{n}(q) \in \mathbb {Z}$ . Show that $\Phi_{n}(q) \mid (q-1) \Longrightarrow n=1$ .

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Problem III.5.2.3

Verify that the cyclotomic polynomial

\Phi _{15}(X)=X^{8}-X^{7}+X^{5}-X^{4}+X^{3}-X+1

considered over the field $\mathbb {F}_{2}$ , is the product of two irreducible polynomials $X^{4}+X^{3}+1$ and $X^{4}+X+1$ . Using this fact, prove the irreducibility of $\Phi_{15}(X)$ over $\mathbb {Q}$ (cf. Exercise 11 from [ВА I, Ch. 6, § 1]).

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Problem III.5.2.4

Verify the following properties of cyclotomic polynomials.

If $p$ is a prime number and $p \mid n$ , then $\Phi_{p n}(X)=\Phi_{n}\left(X^{p}\right)$ ; if, however, $p \nmid n$ , then $\Phi_{p n}(X)=\Phi_{n}\left(X^{p}\right) / \Phi_{n}(X)$ .

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Problem III.5.2.5

Starting from the chain of natural inclusions

G F(p) \subset G F\left(p^{2 !}\right) \subset G F\left(p^{3 !}\right) \subset \ldots ,

introduce the so-called limit field $\Omega_{p}=G F\left(p^{\infty }\right)$ , setting

\alpha \in \Omega _{p} \Longleftrightarrow \left\{ \alpha \in G F\left(p^{n !}\right) \text{ for sufficiently large } n\right\} \text{. }

Relying on the basic properties of finite fields, prove that $\Omega_{p}$ is an algebraically closed field. In this way one obtains, together with the field of complex numbers $\mathbb {C}$ , examples of algebraically closed fields of any characteristic.

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Problem III.5.2.6

Let $q=p^{n}$ . Show that for $p=2$ all elements of the field $\mathbb {F}_{q}$ are squares, while for $p>2$ the squares of the group $\mathbb {F}_{q}^{*}$ form in it a subgroup $\mathbb {F}_{q}^{* 2}$ of index 2, with $\mathbb {F}_{q}^{* 2}=\operatorname {Ker}\left(t \mapsto t^{(q-1) / 2}\right)$ .

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Problem III.5.2.7

Let $\mathbb {F}_{q}$ be a finite field with an odd number $q=p^{n}$ of elements. If $q$ is not equal to 3 or 5, then on the "circle" $x^{2}+y^{2}=1$ there is a point with coordinates $x, y \in \mathbb {F}_{q}^{*}$ . Prove this statement for $p>5$ .

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Problem III.5.2.8

Is every primitive element of the field $\mathbb {F}_{q}$ a generator of the multiplicative group $\mathbb {F}_{q}^{*}$ ?

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Problem III.5.2.9

Let $A(q)=\operatorname {Ass}_{F}\left(X_{1}, \ldots , X_{q}\right)$ be the free associative algebra over a field $F$ , generated by $q$ free generators — noncommuting variables $X_{1}, \ldots , X_{q}$ . Setting

A_{m}(q)= \left\langle X_{i_{1}} X_{i_{2}} \ldots X_{i_{m}} \mid 1 \leqslant i_{j} \leqslant q\right\rangle _{F}, \quad \operatorname {dim} A_{m}(q)=q^{m}

we note that $A(q)$ is a graded algebra:

A(q)=F \cdot 1 \oplus A_{1}(q) \oplus A_{2}(q) \oplus A_{3}(q) \oplus \ldots

$A(q)$ contains the free Lie algebra $L(q)=\operatorname {Lie}\left(X_{1}, \ldots , X_{q}\right)$ with the same free generators and the commutator operation $[U V]=U V-V U$ . The algebra $L(q)$ is also graded:

L(q)=L_{1}(q) \oplus L_{2}(q) \oplus L_{3}(q) \oplus \ldots

where

L_{1}(q)= \left\langle X_{1}, \ldots , X_{q}\right\rangle _{F}, \quad L_{2}(q)= \left\langle \left[X_{i}, X_{j}\right] \mid i<j\right\rangle _{F}, \quad \ldots

Using the Jacobi identity, verify that

L_{3}(q)= \left\langle \left[\left[X_{i}, X_{j}\right], X_{k}\right]\right\rangle , \quad i<j, k \leqslant j ; \quad \operatorname {dim} L_{3}(q)=\frac{1}{3}\left(q^{3}-q\right)

In fact, the general Witt formula holds

\operatorname {dim} L_{m}(q)=\Psi _{m}(q)=\frac{1}{m} \sum _{d \mid m} \mu \left(\frac{m}{d}\right) q^{d}

coinciding exactly with formula (12). The essential difference is only that in (12') $q$ is an arbitrary natural number, while in (12) it is a power of a prime number.

The number of necklaces from Problem 2 in the preamble to Ch. 3 is expressed by the same formula.

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