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tags: mth350
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# MTH 350: Modern Algebra Problem List

:::info
Problems with :ballot_box_with_check: next to them have been completed in Daily Prep, assigned as Homework, or submitted to the Class Journal.
:::

:::warning
**Using these Problems to justify steps in other proofs:**
All Problems that are phrased as "Show that..." or "Prove that..." are true statements and can be used to justify steps in other proofs, *as long as the Problem you're using doesn't depend on the theorem you are proving*. For example if you need Problem 2.11 to prove Problem 2.12, then you can't use Problem 2.12 to justify a step in the proof of Problem 2.11 because that's [circular logic](https://examples.yourdictionary.com/circular-reasoning-fallacy-examples.html). Generally speaking **a problem cannot be used in proofs of problems that precede it**, although there may be exceptions.

Other problems (for example the ones that say "Prove or disprove") are not necessarily true statements, so don't use them in other proofs until someone has proven them.
:::

## Fields

:ballot_box_with_check: **Problem 2.1:** Show that for every nonzero complex number $a+bi$, there is a complex number $c + di$ such that $(a+bi)(c+di) = 1$.

:ballot_box_with_check: **Problem 2.2:** Determine which of the following are fields:
(a) $\mathbb{Z}_6$ with addition and multiplication mod 6
(b) $\mathbb{Z}_5$ with addition and multiplication mod 5
(c) $M_2(\mathbb{R})$, the set of all $2 \times 2$ matrices with real number entries, with standard matrix addition and multiplication

**Problem 2.3:** Let $F$ be a field. Prove that:
(a) For all $x \in F$, $(-1_F)x = -x$. (Note: "$-1_F$" means the additive inverse of the multiplicative identity of $F$. It does **not** refer to the literal number $-1$.)
(b) For all $x \in F$, $-(-x) = x$.
(c) For all $x,y \in F$, $(-x)y = - (xy)$.
Side note to think about: Which field axioms did you use?

**Problem 2.4:** (*Theorem 2.15*) Let $F$ be a field. Prove the following:
(a) The additive identify $0_F$ is unique.
(b) For all $a \in F$, $a \cdot 0_F = 0_F \cdot a = 0_F$. (Note: "$0_F$" means the additive identity of $F$. It does **not** refer to the literal number $0$.)
(c) Additive inverses are unique.
(d) The multiplicative identity $1_F$ is unique.
(e) Multiplicative inverses are unique.
(f) $(-1_F)(-1_F) = 1_F$. (Note: "$-1_F$" means the additive inverse of the multiplicative identity of $F$. It does **not** refer to the literal number $-1$.)
**Note:** You may use the results of Problem 2.3 if they help, even if nobody has proven those yet.

**Problem 2.5:** Prove that $\mathbb{Z}_p$ is a field if and only if $p$ is a prime number. (Note: This is an "if and only if" statement so there are two directions to prove.)

**Problem 2.6:** Prove or disprove that there exists a field with eight elements in it. (Note, if such a field exists, it won't contradict Problem 2.5 --- it would just mean that this field can't be $\mathbb{Z}_8$.)

## Rings

:ballot_box_with_check: **Problem 2.7:** We say that a property of a set $S$ is **inherited** if for every nonempty subset $T$ of $S$ also has that property. For example, in the natural numbers, the property that every element is positive, is inherited.
(a) Prove that for all rings, the commutative property of addition, associative properties for both addition and multiplication, and distributive properties are inherited.
(b) Prove that for all commutative rings, the commutative property of multiplication is inherited.
(c) Give an example that shows that for rings, the property of having an additive identity is *not* inherited; and an example that shows that for rings, the property that every element has an additive inverse is not inherited.

**Problem 2.8:** (*Theorem 2.2.6*, a.k.a. "The Subring Test") Let $R$ be a ring and $S$ a subset of $R$. Then $S$ is a subring if and only if:
1. $S \neq \emptyset$;
2. $S$ is closed under multiplication; and
3. $S$ is closed under subtraction.

**Problem 2.9:** Prove that if $R$ is any ring, then $R$ is a subring of $R$, and so is $\{0_R\}$, the set consisting of just the additive identity of $R$.

:ballot_box_with_check: **Problem 2.10:** Prove or disprove:
(a) If $R$ is a ring with identity $1_R$ and $S$ is a subring of $R$, then $S$ also has an identity.
(b) If $R$ is a ring with identity $1_R$, $S$ is a subring of $R$, and $S$ has an identity, then the identity of $S$ is $1_R$.
(c) If $R$ is a commutative ring and $S$ is a subring of $R$, then $S$ is also commutative.
(d) If $R$ is a *non*-commutative ring and $S$ is a subring of $R$, then $S$ is also non-commutative.

:ballot_box_with_check: **Problem 2.11:**
* Find all the elements of $\mathbb{Z}_6^\times$, and then for each $x \in \mathbb{Z}_6$, find all the associates of $x$. (Some elements may not have any associates.)
* Find all the elements of $\mathbb{Z}^\times$. For 4-5 elements of $\mathbb{Z}$, find all the associates of each. Then make and prove a conjecture that completes the statement: *For every $a \in \mathbb{Z}$, the associates of $a$ are ____*.

:ballot_box_with_check: **Problem 2.12:** (*Theorem 2.2.8*, rephrased) If $R$ is a commutative ring with identity and every nonzero element is a unit, then $R$ is a field.

:ballot_box_with_check: **Problem 2.13:** (*Theorem 2.2.12*) Let $R$ be a ring and suppose $a,b \in R$ such that $ab$ is a zero divisor. Then either $a$ or $b$ is a zero divisor.

:ballot_box_with_check: **Problem 2.14**: (*Theorem 2.2.13*) Let $R$ be a ring and $u \in R^\times$. Then $u$ is not a zero divisor.

:ballot_box_with_check: **Problem 2.15**: (*Theorem 2.2.15*) Every field is an integral domain. Also, give an example where the converse fails --- an integral domain that is not a field.

:ballot_box_with_check: **Problem 2.16**: (*Theorem 2.2.16*) Let $m > 1$ and $R = \mathbb{Z}_m$. Then $R$ is a field if and only if $R$ is an integral domain.

**Problem 2.17**: (*Theorem 2.2.17*) If $R$ is an integral domain and $S$ is a subring of $R$ with identity $1_S = 1_R$, then $S$ is an integral domain.

**Problem 2.18**: Give an example where Theorem 2.2.17 fails if $1_S \neq 1_R$.

**Problem 2.19:** (*Theorem 2.2.18*) If $R$ is an integral domain, then so is $R[x]$.

**Problem 2.20:** (*Theorem 2.2.20*) Let $R$ be an integral domain, and let $p(x), q(x) \in R[x]$ be nonero polynomials. Then $\deg(p(x)q(x)) = \deg(p(x)) + \deg(q(x))$.

:ballot_box_with_check: **Problem 2.21:** Give an example where Theorem 2.2.20 fails if $R$ is not an integral domain.

**Problem 2.22:** (*Investigation 2.2.18*) Suppose $R$ is an integral domain. Characterize all the units of $R[x]$ and prove that your result is correct.

## Divisibility in Integral Domains

:ballot_box_with_check: **Problem 2.23:** (*Theorem 2.3.5*) Let $R$ be an integral domain. If $a \in R$ is prime, then $a$ is irreducible.

**Problem 2.24:** (*Theorem 2.3.6*) Every irreducible in $\mathbb{Z}$ is prime.

## Principal ideals and Euclidean domains

:ballot_box_with_check: **Problem 2.25:** (*Theorem 2.4.2*) Let $R$ be a ring. Then $R$ and $\{0_R\}$ are ideals of $R$.

:ballot_box_with_check: **Problem 2.26:** (*Theorem 2.4.3*) All ideals are subrings.

**Problem 2.27:** (*Theorem 2.4.4*) Let $R$ be a ring and $I$ an ideal of $R$. Then $I = R$ if and only if $I$ contains a unit of $R$.

:ballot_box_with_check: **Problem 2.28:** Let $R$ be a commutative ring with identity and let $a \in R$. Then $\langle a \rangle = \{ra \, : \, r \in R \}$ is an ideal of $R$.

:ballot_box_with_check: **Problem 2.29:** (*Theorem 2.4.6*) Let $R$ be a ring and $a \in R$. Then $\langle a \rangle = \langle ua \rangle$ where $u$ is any unit in $R$.

**Problem 2.30:** (*Theorem 2.4.8*) The ring $\mathbb{Z}$ is a principal ideal domain.

**Problem 2.31**: (*Theorem 2.4.9*, Generalized Bezout's Identity) Let $R$ be a principal ideal domain and $x,y \in R$ be not both zero. Let $I = \{xm + yn \, : \, m,n \in R\}$. Then:
(1) $I$ is an ideal; and
(2) $I = \langle d \rangle$ where $d$ is a greatest common divisor of $x$ and $y$. In particular, there exist $s,t \in R$ such that $d = xs + yt$.

**Problem 2.32:** (*Theorem 2.4.11*) The field $\mathbb{Q}$ is a Euclidean domain with norm function $\delta (x) = 0$ for all $x \in \mathbb{Q}$.

**Problem 2.33:** (*Lemma 2.4.12*) Let $F$ be a field and $S \subseteq F$ containing a nonzero polynomial. Prove there exists a polynomial $f \in S$ such that $\deg(f) \leq \deg(g)$ for all nonzero $g \in S$.

**Problem 2.34:** (*Lemma 2.4.13*) Let $F$ be a field and $f(x), g(x) \in F[x]$ with $g(x) \neq 0$. If $\deg(f(x)) \geq \deg(g(x)) > 0$ and $f(x) = a_0 + a_1x + \cdots + a_mx^m$ and $g(x) = b_0 + b_1x + \cdots + b_nx^n$, define $h(x) = f(x) - a_mb^{-1}_n x^{m-n}g(x)$. Then $\deg(h(x)) < \deg(f(x))$.

**Problem 2.35:** (Polynomial Division Algorithm, Existence) Let $F$ be a field and $f(x), g(x) \in F[x]$ with $g(x) \neq 0$. Then there exist polynomials $q(x), r(x) \in F[x]$ such that
$$f(x) = g(x) q(x) + r(x)$$
where $\deg(r(x)) < \deg(g(x))$.

**Problem 2.36:** (Polynomial Division Algorithm, Uniqueness) The polynomials $q(x)$ and $r(x)$ from Problem 2.35 are the only ones that satisfy the two properties listed.

**Problem 2.37:** Prove or disprove that the Polynomial Division Algorithm holds for $\mathbb{Z}[x]$, the ring of all polynomials with integer coefficients.

**Problem 2.38:** (*Theorem 2.4.15*) If $F$ is a field, then $F[x]$ is a principal ideal domain.

**Problem 2.39:** Prove or disprove that if $F$ is a field, then $F[x]$ is a Euclidean domain.

**Problem 2.40:** (*Theorem 2.4.16*) Prove that every Euclidean domain is a principal ideal domain.