1.15 Exercises

Let $E={\mathbb{Q}}[\alpha]$, where $\alpha^{3}-\alpha^{2}+\alpha+2=0$. Express $(\alpha^{2}+\alpha+1)(\alpha^{2}-\alpha)$ and$(\alpha-1)^{-1}$ in the form $a\alpha^{2}+b\alpha+c$ with $a,b,c\in{\mathbb{Q}}$.

Determine $[{\mathbb{Q}}(\sqrt{2},\sqrt{3})\colon{\mathbb{Q}}]$.

Let $F$ be a field, and let $f(X)\in F[X]$.

1. (a)
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   For every $a\in F$, show that there is a polynomial $q(X)\in F[X]$ such that

   $$f(X)=q(X)(X-a)+f(a).$$
2. (b)
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   Deduce that $f(a)=0$ if and only if $(X-a)|f(X)$.

3. (c)
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   Deduce that $f(X)$ can have at most $\deg f$ roots.

4. (d)
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   Let $G$ be a finite abelian group. If $G$ has at most $m$ elements of order dividing $m$ for each divisor $m$ of $(G\colon 1)$, show that $G$ is cyclic.

5. (e)
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   Deduce that every finite subgroup of $F^{\times}$, $F$ a field, is cyclic.

Show that with straight-edge, compass, and angle-trisector, it is possible to construct a regular $7$-gon.

Let $f(X)$ be an irreducible polynomial over $F$ of degree $n$, and let $E$ be a field extension of $F$ with $[E:F]=m$. If $\gcd(m,n)=1$, show that $f$ is irreducible over $E$.

Show that there does not exist a polynomial $f(X)\in\mathbb{Z}{}[X]$ of degree $>1$ that is irreducible modulo $p$ for all primes $p$.

Let $\alpha=\sqrt[3]{2}$, and let $R$ be the set of complex numbers of the form $a+b\alpha+c\alpha^{2}$ with $a,b,c\in\mathbb{Q}{}$. Show that $R$ is a field.