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Problem 79 (In A Euclidean Domain All Units Have The Same Measure)
Suppose $D$ is a Euclidean domain with measure $d$. Let $u$ be a unit. Prove that that $d(u)=d(1)$, in other words prove that all units must have the exact same measure.
Problem 80 (In A Euclidean Domain Associates Have The Same Measure)
Suppose $D$ is a Euclidean domain with measure $d$. Suppose that $a$ and $b$ are nonzero associates, in other words assume that $a=ub$ for some unit $u$. Prove that $d(a)=d(b)$.
Problem 81 (Describing the Extension Field $\mathbb{Q}(\pi)$ of $\mathbb{Q}$)
Consider the field $\mathbb{Q}$ and the element $\pi$. It is known that the element $\pi$ is not the zero of any polynomial in $\mathbb{Q}[x]$. However, we also know that $\pi\in \mathbb{R}$, which is an extension field of $\mathbb{Q}$. This means we can talk about $\mathbb{Q}(\pi)$, the smallest subfield of $\mathbb{R}$ that contains both $\mathbb{Q}$ and $\pi$. Describe the elements of $\mathbb{Q}$ in a constructive way. In other words, give a way to obtain all elements of $\mathbb{Q}(\pi)$ by giving a way to describe any element of this field.
The derivative of a polynomial can help us understand the next two problems.
Problem 82 (Irreducibles Over A Field Of Characteristic Zero Have No Repeated Zeros)
Let $F$ be a field of characteristic zero and $f(x)\in F[x]$ be irreducible over $F$. Suppose that $E$ is a splitting field for $f(x)$, so that we can write $f(x) = c(x-a_1)(x-a_2)\cdots(x-a_n)$ for $c\in F$ and $a_1, \ldots, a_n\in E$. Show that $a_i\neq a_j$ if $i\neq j$, in other words show that $f$ has no zero of multiplicity greater than one.
Problem 83 (Irreducibles Over A Field Of Characteristic $p$ May Have Repeated Zeros)
Let $F$ be a field of characteristic $p$ and $f(x)\in F[x]$ be irreducible over $F$. Suppose that $E$ is a splitting field for $f(x)$, so that we can write $f(x) = c(x-a_1)(x-a_2)\cdots(x-a_n)$ for $c\in F$ and $a_1, \ldots, a_n\in E$.
- Show that $f(x)$ might have a zero with multiplicity greater than one (give an example of such a polynomial).
- If $f(x)$ has a multiple zero, then what can we say about the coefficients of $f(x)$?
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For more problems, see AllProblems