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Problem 69 (Showing The Existence of a Field Extension Generated by a subset)
Suppose that $E$ is an extension field of the field $F$. Let $S$ be a subset of $E$.
- Prove that there is a subfield of $E$ that contains both $F$ and $S$.
- Prove that the intersection of all subfields of $E$ that contain both $F$ and $S$ is a subfield of $E$.
Problem 70 (Factoring By An Irreducible Gives A Simple Extension)
Let $F$ be a field and $p(x)\in F[x]$ be irreducible over $F$. If $a$ is a zero of $p(x)$ in some extension $E$ of $F$, then $F(a)$ is isomorphic to $F[x]/\left<p(x)\right>$. Furthermore, if $\deg p(x) = n$, then every member of $F(a)$ can be uniquely expressed in the form $c_{n-1}a^{n-1} + c_{n-2}a^{n-2} + \cdots + c_{1}a + c_{0}$ where $c_i\in F$. (In other words, the set $\left\{1, a, a^2, \ldots, a^{n-1}\right\}$ is a basis for $F(a)$ over $F$).
Problem 71 (Simple Extensions For The Same Polynomial Are Isomorphic)
Let $F$ be a field and let $p(x)\in F[x]$ be irreducible over $F$. If $a$ is a zero of $p(x)$ in some extension $E$ of $F$ and $b$ is a zero of $p(x)$ in some extension $E'$ of $F$, then the fields $F(a)$ and $F(b)$ are isomorphic.
Problem 72 (Irreducible Implies Prime in a PID)
We've already shown that every prime is irreducible. The converse is not always true. However, suppose that $D$ is a principle ideal domain. Prove that if an element $a$ is irreducible then it is prime.
Problem 73 (Unique Factorization In A PID)
Suppose that $D$ is a principal ideal domain. Suppose that $f=p_1p_2\cdots p_m = q_1q_2\cdots q_n,$ where each $p_i$ and $q_i$ is irreducible over $D$. Prove that $n=m$ and after rearranging, for each $i$ we have $p_i=u_iq_i$ for some unit $i$. In other words, prove that $D$ uniquely factors as a product of irreducibles.
We have now shown that in every principle ideal domain, every nonzero nonunit can be written as a product of irreducibles (we'll call this a factorization), and that any two factorization are essentially the same (after rearrangement and multiplying by units). Any integral domain that satisfies this property is called a unique factorization domain.
Definition (Unique Factorization Domain)
Let $D$ be an integral domain. We say that $D$ is a unique factorization domain if the following two properties hold.
- If $d\in D$ is not a unit, and not zero, then we can write $d$ as a product of irreducibles over $D$.
- If we have written $d$ as a product of irreducibles over $D$ in two ways, say $d=p_1p_2\cdots p_n$ and $d=q_1q_2\cdots q_m$, then $n=m$ and after rearranging we have for each $i$ that $p_i=u_iq_i$ for some unit $u_i$.
Problem 74 (Existence Of A Splitting Field)
Let $F$ be a field and $f(x)\in F[x]$ have positive degree. We have already shown that there exists an extension field $E$ of $F$ in which $f(x)$ has a zero $a$, in other words we can write $f(x) = (x-a)g(x)$ where $a,g(x)\in E[x]$ (by the Factor Theorem). Prove that there exists an extension field $E'$ of $F$ in which $f(x)$ can be written as a product of linear factors, i.e. we can write $f(x) = (x-a_1)(x-a_2)\cdots (x-a_n)g$ where $a_1,a_2,\ldots, a_n,g\in E'$ and also $g$ is a unit.
Definition.Splitting Field of $f(x)$ over $F$
Look this on up in the text
The next problem should follow immediately from a fact that we have already shown.
Problem 75 (Polynomial Rings Over A Field Are Unique Factorization Domains)
Prove that $F[x]$ is a unique factorization domain if $F$ is a field.
What was the key that made this all work? What is the key to proving that $F[x]$ is a principle ideal domain? The key is the Division Algorithm, which is also called the Euclidean Algorithm. If an integral domain has something similar to the division algorithm, then we'll call it a Euclidean domain.
Definition.Euclidean Domain
Please look this one up the book.
Take a look at the proof we used to show that $F[x]$ is a PID whenever $F$ is a field. The key to this proof was the division algorithm. You should be able to generalize this proof to show that any Euclidean domain is a principle ideal domain.
Problem 76 (Euclidean Domains Are PIDs)
Prove that a Euclidean domain is a principle ideal domain. Prove also that a Euclidean domain is a unique factorization domain.
Problem 77 (Both Z And FX Are Euclidean Domains)
Prove that $Z$ is a Euclidean domain with function $d(a)=|a|$. Prove that $F[x]$ is a Euclidean domain for a field $F$ with $d(f(x))=\deg f(x)$.
Problem 78 (The Gaussian Integers Is A Euclidean Domain)
Prove that $Z[i]$ is a Euclidean domain, using the function $d(a+bi)=a^2+b^2$.
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 $f(x)$?
For more problems, see AllProblems