2.4 Exercises

Let $F$ be a field of characteristic $\neq 2$.

1. (a)
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   Let $E$ be quadratic extension of $F$; show that

   $$S(E)=\{a\in F^{\times}\mid a\text{{\ }is a square in{\ }}E\}$$

   is a subgroup of $F^{\times}$ containing $F^{\times 2}$.

2. (b)
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   Let $E$ and $E^{\prime}$ be quadratic extensions of $F$; show that there exists an $F$-isomorphism $\varphi\colon E\rightarrow E^{\prime}$ if and only if $S(E)=S(E^{\prime})$.

3. (c)
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   Show that there is an infinite sequence of fields $E_{1},E_{2},\ldots$ with $E_{i}$ a quadratic extension of ${\mathbb{Q}}$ such that $E_{i}$ is not isomorphic to $E_{j}$ for $i\neq j$.

4. (d)
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   Let $p$ be an odd prime. Show that, up to isomorphism, there is exactly one field with $p^{2}$ elements.

(a) Let $F$ be a field of characteristic $p$. Show that if $X^{p}-X-a$ is reducible in $F[X]$, then it splits into distinct factors in $F[X]$.

(b) For every prime $p$, show that $X^{p}-X-1$ is irreducible in ${\mathbb{Q}}[X]$.

Construct a splitting field for $X^{5}-2$ over ${\mathbb{Q}}$. What is its degree over ${\mathbb{Q}}$?

Find a splitting field of $X^{p^{m}}-1\in\mathbb{F}_{p}[X]$. What is its degree over $\mathbb{F}_{p}$?

Let $f\in F[X]$, where $F$ is a field of characteristic $0$. Let $d(X)=\gcd(f,f^{\prime})$. Show that $g(X)=f(X)d(X)^{-1}$ has the same roots as $f(X)$, and these are all simple roots of $g(X)$.

Let $f(X)$ be an irreducible polynomial in $F[X]$, where $F$ has characteristic $p$. Show that $f(X)$ can be written $f(X)=g(X^{p^{e}})$ where $g(X)$ is irreducible and separable. Deduce that every root of $f(X)$ has the same multiplicity $p^{e}$ in any splitting field.