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\noindent{\bf Math 431 homework}\hfill Due Friday, September 14, 2001

\ex 
Find the product of all nonzero elements of a finite field with~$q$ elements.

\ex
If $E/F$ is a finite extension of fields, prove that every element of
$E$ is algebraic over~$F$.

\ex
Prove or disprove: the field $\Q(\sqrt{5})$ is isomorphic to the
field $\Q(\sqrt{7})$.

\ex
Prove that if $E$ is a splitting field of a polynomial $f(x)$ of 
degree~$n$ over a field~$F$, then $[E:F] \le n!$.

\ex
Let $F$ be a root field over $\Q$ of the polynomial $f(x) = x^4-2$.
Factor $f(x)$ in $F[x]$.

\ex Same as the previous question, except with $f(x) = x^4 +1$.

\ex
Find a field $F$ inside~$\R$ of as small a degree over~$\Q$ as possible 
such that there is a regular pentagon in the plane whose vertices  $(x,y)$
have their coordinates $x,y$ lying in~$F$.

\ex
Let $a$ and $b$ be integers and $p$ a prime.  Show that
$${pa \choose pb} \equiv {a \choose b} \bmod p.$$

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