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\noindent{\Large{\textsc{Modern Algebra II\hfill Assignment 10.3  \\ Fall 2019 \hfill Due November 20}}}

\large

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\begin{exercise}
Let $K/F$ be fields and suppose $\alpha \in K$ is algebraic over $F$ with minimal polynomial $p \in F[X]$.  If $q \in F[X]$ is monic, irreducible and satisfies $q(\alpha) = 0$, show that $p = q$.  
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\begin{exercise}
Let $F$ be a field and let $f, g \in F[X]$ be nonzero.  Suppose that $K$ is an extension of $F$ and there is an $\alpha \in K$ so that $f(\alpha) = g(\alpha) = 0$.  Prove that $f$ and $g$ have a nontrivial common factor in $F[X]$.\footnote{The polynomial $X- \alpha$ doesn't count as a common factor:  it divides both $f$ and $g$, but not necessarily over $F$.}  Conclude that two polynomials over $F$ have a nontrivial common factor over $F$ if and only if they share a root in an extension of $F$.
\end{exercise}

\begin{exercise}
Let $\alpha$ be a root of $X^3 + X^2 + 1$ in some extension of $\mathbb{F}_2 = \Z/2\Z$.  What is the degree of the minimal polynomial of $\alpha$ over $\mathbb{F}_2$?
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\begin{exercise}
Lang, Exercise VII.1.23.
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