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Problem 64 (Ascending Chains of Ideals In a PID Are Finite)

Suppose that $R$ is a ring, and that $A_1\subseteq A_2\subseteq A_3\subseteq \cdots$ is nested sequence of ideals of $R$.

  1. Prove that the union $\ds \bigcup_{n=1}^{\infty} A_n$ is an ideal of $R$.
  2. If $R$ is a principle ideal domain, prove that there exists an integer $k$ so that $A_k=A_n$ for every $n\geq k$. In other words, prove that every ascending chain of ideals in a principle ideal domain contain finitely many different ideals.
  3. Challenge: Give an example of a ring $R$ and an infinite ascending chain of ideals so that no two ideals are equal.

You can present in class even if you don't get the third part.

Problem 65 (Every Element is a Product of Irreducibles in a PID)

Suppose that $D$ is a principle ideal domain with $a\in D$ where $a\neq 0$ is not a unit. Prove that $a$ can be written as the product of irreducibles.

Hint: You'll want to use the previous problem quite a bit. If $a$ is irreducible, then you're done. You'll probably want to start by showing that you can write $a=pc$ for some irreducible $p$ (in other words, you can factor off an irreducible. How will you use the previous problem? If you can't factor off an irreducible, then you can write $a=a_1b_1$ for two non unit non irreducible terms. Why do we know $\langle a\rangle\subseteq\langle a_1\rangle$? Now repeat this process to get $\langle a\rangle\subseteq\langle a_1\rangle\subseteq\langle a_2\rangle$ and so on. Once you've gotten $a=pc$ for some irreducible $p$, what connection is there between $\langle a\rangle$ and $\langle c\rangle$?

Problem 66 (Every Polynomial Has A Root In Some Extension Field)

Let $F$ be a field and $f(x)\in F[x]$ be nonzero and not a unit. Prove that there exists an extension field $E$ in which $f(x)$ has a zero.

Definition.The Derivative Of A Polynomial Over a Ring

Let $R$ be a ring and $f(x)\in R[x]$. If $f(x)=0$, then we define the derivative of $f(x)$ to be zero. Otherwise, we define the derivative of $f(x) = a_nx^n+\cdots +a_1x+a_0$ where $a_n\neq 0$ to be $$f'(x) = n\cdot a_nx^{n-1}+(n-1)a_{n-1}x^{n-2}+\cdots + 2a_2 x + a_1.$$

Problem 67 (The Sum and Product Rule for Derivatives)

Suppose that $F$ is a field and that $f(x),g(x)\in F[x]$.

  1. Prove the sum rule, namely that $(f+g)'(x)=f'(x)+g'(x)$.
  2. Prove the product rule, namely that $(fg)'(x)=f'(x)g(x)+f(x)g'(x)$.
  3. What properties of the field $F$ were needed to complete your proofs? State a stronger theorem that requires less assumptions.

Problem 68 (Multiple Zeros and The Derivative Proof)

Suppose that $F$ is field and $f(x)\in F[x]$. Prove that $f(x)$ has a zero of multiplicity greater than one in some extension field $E$ if and only if $f$ and $f'$ have a common factor in $F[x]$.

Defintion.Field extensions generated by subsets and Simple Extensions

Suppose that $E$ is a field extension of $F$ and $S\subset E$.

  1. We'll use the notation $F(S)$ to represent the smallest subfield of $E$ that contains both $F$ and $S$.
  2. If $S=\{a\}$ then we call $F(a)$ a simple extension.

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 $E$.

  1. Prove that there is a subfield of $E$ that contains both $F$ and $S$.
  2. 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.

For more problems, see AllProblems