Ring and Field Theory

This chapter provides a rigorous introduction to Ring and Field Theory, fundamental concepts within abstract algebra. A thorough understanding of rings, ideals, and fields, along with their associated properties, is essential for the CUET PG MA examination, as these topics frequently appear in both theoretical and problem-solving contexts.

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Chapter Contents

| # | Topic |
|---|-------|
| 1 | Rings and Subrings |
| 2 | Ideals |
| 3 | Prime and Maximal Ideals |
| 4 | Fields and Quotient Fields |

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We begin with Rings and Subrings.

Part 1: Rings and Subrings

This section provides a concise overview of rings, subrings, and their fundamental properties. It covers definitions, key theorems, and common examples essential for quick revision, focusing on distinguishing between different types of rings and identifying subrings.

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Formula Sheet

| # | Concept | Formula | When to Use |
|---|---------|---------|-------------|
| 1 | Ring | $(R, +, \cdot)$ is a ring if:
(1) $(R, +)$ is an Abelian group.
(2) $(R, \cdot)$ is a semigroup (associative multiplication).
(3) Distributive laws hold: $a \cdot (b+c) = a \cdot b + a \cdot c$ and $(a+b) \cdot c = a \cdot c + b \cdot c$ | Basic structure definition |
| 2 | Commutative Ring | $a \cdot b = b \cdot a$ for all $a, b \in R$ | When multiplication is commutative |
| 3 | Ring with Unity | There exists $1_R \in R$ such that $1_R \cdot a = a \cdot 1_R = a$ for all $a \in R$ | When a multiplicative identity exists |
| 4 | Zero Divisor | A non-zero element $a \in R$ such that there exists a non-zero $b \in R$ with $a \cdot b = 0$ or $b \cdot a = 0$ | To check for presence of zero divisors |
| 5 | Integral Domain (ID) | A commutative ring with unity $1_R \neq 0$ that has no zero divisors | To classify rings like $\mathbb{Z}$ or $\mathbb{Z}_p$ (p prime) |
| 6 | Field | A commutative ring with unity $1_R \neq 0$ where every non-zero element has a multiplicative inverse | To classify rings like $\mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_p$ (p prime) |
| 7 | Division Ring (Skew Field) | A ring with unity $1_R \neq 0$ where every non-zero element has a multiplicative inverse (multiplication not necessarily commutative) | Generalization of field (e.g., Quaternions) |
| 8 | Subring Test | A non-empty subset $S$ of a ring $R$ is a subring if:
(1) $S$ is closed under subtraction: $a,b \in S \implies a-b \in S$.
(2) $S$ is closed under multiplication: $a,b \in S \implies a \cdot b \in S$. | To verify if a subset is a subring |
| 9 | Characteristic of a Ring | Smallest positive integer $n$ such that $n \cdot a = 0$ for all $a \in R$. If no such $n$ exists, characteristic is 0. | For finite rings or rings like $\mathbb{Z}_n$. For $R$ with unity, it's the smallest $n$ s.t. $n \cdot 1_R = 0$. |
| 10 | Boolean Ring | A ring where every element is idempotent ($a^2 = a$ for all $a \in R$) | Properties: always commutative, characteristic 2 |
| 11 | Polynomial Ring | $R[x] = \{a_n x^n + \dots + a_1 x + a_0 \mid a_i \in R, n \ge 0\}$ | Ring formed by polynomials with coefficients in $R$ |

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Key Definitions

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Must Remember

Hierarchy of Rings: Field $\implies$ Integral Domain $\implies$ Commutative Ring with Unity.

Finite Integral Domain: Every finite integral domain is a field.

$\mathbb{Z}_n$ Properties: The ring $\mathbb{Z}_n$ (integers modulo $n$) is an integral domain if and only if $n$ is a prime number. It is a field if and only if $n$ is a prime number.

Subring of $\mathbb{Z}$: For any positive integer $n$, the set $n\mathbb{Z} = \{nk \mid k \in \mathbb{Z}\}$ is a subring of $\mathbb{Z}$.

Idempotent Rings (Boolean Rings): A ring where every element is idempotent ($a^2=a$) is always commutative and has characteristic 2 (i.e., $a+a=0$ for all $a$).

Cancellation Law: The cancellation laws ($ab=ac \implies b=c$ and $ba=ca \implies b=c$) hold in a ring $R$ if and only if $R$ has no zero divisors.

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Common Mistakes

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Previous Year Questions

type="MCQ" question="[Year: 2023] Which of the following rings is not an integral domain?" options=["$\mathbb{Z}_{100}$","$\mathbb{Z}_{102}$","$\mathbb{Z}_{113}$","$\mathbb{Z}_{153}$"] answer="$\mathbb{Z}_{100}$" hint="Recall the condition for $\mathbb{Z}_n$ to be an integral domain." solution="A ring $\mathbb{Z}_n$ is an integral domain if and only if $n$ is a prime number.

• $100 = 10 \times 10$, so $\mathbb{Z}_{100}$ is not an integral domain (e.g., $10 \cdot 10 = 0 \pmod{100}$).


• $102 = 2 \times 51$, so $\mathbb{Z}_{102}$ is not an integral domain (e.g., $2 \cdot 51 = 0 \pmod{102}$).


• $113$ is a prime number, so $\mathbb{Z}_{113}$ is an integral domain.


• $153 = 3 \times 51$, so $\mathbb{Z}_{153}$ is not an integral domain (e.g., $3 \cdot 51 = 0 \pmod{153}$).

The question asks for a ring that is not an integral domain. Multiple options fit this criterion. As an MCQ, typically one answer is expected. We choose $\mathbb{Z}_{100}$ as a valid example."

type="MCQ" question="[Year: 2025] A ring $(R, +, \cdot)$, where all elements are idempotent is always:" options=["a commutative ring","not an integral domain","a field","an integral domain with unity"] answer="a commutative ring" hint="Consider the property $(a+b)^2 = a+b$ for all $a,b \in R$." solution="Let $R$ be a ring where every element is idempotent, i.e., $x^2 = x$ for all $x \in R$.

Commutativity: For any $a,b \in R$, we have $(a+b)^2 = a+b$. Expanding this, $a^2 + ab + ba + b^2 = a+b$. Since $a^2=a$ and $b^2=b$, we get $a + ab + ba + b = a+b$. This implies $ab+ba = 0$, or $ab = -ba$.

Also, for any $a \in R$,



Since $a^2=a$,


This means $a+a=0$, or $a=-a$ for all $a \in R$.
Substituting $a=-a$ into $ab=-ba$, we get $ab=ba$. Thus, the ring is commutative.

Integral Domain/Field: An idempotent ring is an integral domain if and only if it has exactly two elements, $\{0,1\}$, which forms $\mathbb{Z}_2$. $\mathbb{Z}_2$ is also a field. However, not all idempotent rings are integral domains (e.g., $\mathbb{Z}_2 \times \mathbb{Z}_2$ is idempotent but has zero divisors like $(1,0)(0,1)=(0,0)$). Therefore, it is not always 'not an integral domain', 'a field', or 'an integral domain with unity'."

type="MCQ" question="[Year: 2022] A. For a matrix A, $A^T A$ is always symmetric.
B. $A + A^T$ is a symmetric matrix.
C. In a group G, if $a, b, c \in G$, are such that $ab = ac$, then $b = c$.
D. In the ring R, for $a, b \in R$, $(a+b)^2 = a^2 + b^2 + ab + ba$.
E. Every integral domain is a field.
Choose the correct answer from the options given below:" options=["A, B, C, D only","A, B, C, D, E only","A, B only","B, C only"] answer="A, B, C, D only" hint="Recall definitions of symmetric matrices, group cancellation, ring multiplication, and the relationship between integral domains and fields." solution="Let's evaluate each statement:
A. For a matrix A,

Thus, $A^T A$ is always symmetric. (True)
B. For a matrix A,

Thus, $A + A^T$ is a symmetric matrix. (True)
C. In a group G, the cancellation law holds. If $ab=ac$, multiplying by $a^{-1}$ on the left gives




(True)
D. In a ring R, multiplication distributes over addition. So,

(True)
E. Every integral domain is a field. This is false. For example, the ring of integers $\mathbb{Z}$ is an integral domain but not a field, because non-zero elements like $2$ do not have multiplicative inverses in $\mathbb{Z}$. (False)
Therefore, statements A, B, C, D are true, and E is false."

type="MCQ" question="[Year: 2023] If $P(x)$ is a polynomial of degree 10 with leading coefficient as 11 and having $(x-1)$, $(x-2)$, $(x-3)$,...$(x-10)$ as factor, then the coefficient of $x^9$ in $P(x)$ is" options=["$-55$","$605$","$-605$","$55$"] answer="$-605$" hint="A polynomial with roots $r_1, r_2, \dots, r_n$ can be written as $P(x) = C(x-r_1)(x-r_2)\dots(x-r_n)$. The coefficient of $x^{n-1}$ is $-C \sum r_i$." solution="The polynomial $P(x)$ has degree 10, leading coefficient 11, and roots $1, 2, \dots, 10$.
So,

To find the coefficient of $x^9$, we use Vieta's formulas. For a monic polynomial $x^n + a_{n-1}x^{n-1} + \dots + a_0$, the coefficient $a_{n-1}$ is the negative sum of its roots.
In our case, the product $(x-1)(x-2)\dots(x-10)$ expands to $x^{10} - (1+2+\dots+10)x^9 + \dots$.
The sum of the roots is

So,

Therefore,

The coefficient of $x^9$ is $11 \times (-55) = -605$."

type="MCQ" question="[Year: 2024] Match List-I with List-II

Choose the correct answer from the options given below:" options=["(A) - (IV), (B) - (II), (C) - (III), (D) - (I)","(A) - (III), (B) - (II), (C) - (I), (D) - (IV)","(A) - (III), (B) - (I), (C) - (IV), (D) - (II)","(A) - (IV), (B) - (II), (C) - (I), (D) - (III)"] answer="(A) - (IV), (B) - (II), (C) - (I), (D) - (III)" hint="Recall definitions of different ring types and properties of given sets." solution="Let's match each item from List-I with its correct description from List-II:
(A). Set of all even integers ($2\mathbb{Z}$):
- This is a commutative ring under standard addition and multiplication.
- It does not contain a multiplicative identity (1 is not an even integer).
- It is not an integral domain (lacks unity) and thus not a field.
- So, (A) matches with (IV) Commutative ring.

(B). Set $\{a + ib: a, b \in \mathbb{Z}\}$ ($\mathbb{Z}[i]$, Gaussian Integers):
- This is a subring of the field of complex numbers $\mathbb{C}$.
- It is a commutative ring with unity ($1 = 1+0i$).
- Since it's a subring of a field, it has no zero divisors.
- Therefore, $\mathbb{Z}[i]$ is an Integral domain.
- It is not a field (e.g., $2$ does not have a multiplicative inverse in $\mathbb{Z}[i]$).
- So, (B) matches with (II) Integral domain.

(C). Set of rational numbers ($\mathbb{Q}$):
- The set of rational numbers forms a field under standard addition and multiplication. Every non-zero rational number has a multiplicative inverse.
- So, (C) matches with (I) Field.

(D). **Set $S = \left\{\begin{bmatrix}0 & x\\0 & y\end{bmatrix} \middle| x,y \in \mathbb{Q}\right\}$:**
- This set forms a ring under matrix addition and multiplication.
Let

Both $A, B \in S$.


- Since $AB \neq BA$, the ring is non-commutative.
- So, (D) matches with (III) Non-Commutative ring.

Combining the matches: (A)-(IV), (B)-(II), (C)-(I), (D)-(III)."

type="MCQ" question="[Year: 2022] Given below are two statements :

Statement I : There exists no integral domain of order 6.

Statement II : The total number of distinct permutations on the set $S = \{1, 2, 3\}$ is 6.

In the light of the above statements, choose the correct answer from the options given below :" options=["Both statement I and statement II are true","Both statement I and statement II are false","Statement I is true but statement II is false","Statement I is false but statement II is true"] answer="Both statement I and statement II are true" hint="Recall the possible orders of finite integral domains and the formula for permutations." solution="Let's analyze each statement:
Statement I: A finite integral domain is always a field. The order of a finite field must be a prime power $p^n$ for some prime $p$ and positive integer $n$. Since $6 = 2 \times 3$ is not a prime power, there exists no field of order 6. Consequently, there exists no integral domain of order 6. (True)
Statement II: The number of distinct permutations on a set with $n$ distinct elements is $n!$. For the set $S = \{1, 2, 3\}$, the number of elements is $n=3$. So, the total number of distinct permutations is $3! = 3 \times 2 \times 1 = 6$. (True)
Both statements are true."

type="MCQ" question="[Year: 2022] Given below are two statements :

Statement I :
For each positive integer n, the set $n\mathbb{Z} = \{0, \pm n, \pm 2n, \pm 3n, ...\}$ is a subring of the ring of integers.

Statement II :
$(\mathbb{Z}_6, +_6, \times_6)$ is an integral domain.

In the light of the above statements, choose the correct answer from the options given below :" options=["Both statement I and statement II are correct","Both statement I and statement II are incorrect","Statement I is correct but statement II is incorrect","Statement I is incorrect but statement II is correct"] answer="Statement I is correct but statement II is incorrect" hint="Apply the subring test for Statement I and the definition of an integral domain for Statement II." solution="Let's evaluate each statement:
Statement I: To check if $n\mathbb{Z}$ is a subring of $\mathbb{Z}$, we use the subring test:

$n\mathbb{Z}$ is non-empty (e.g., $0 \in n\mathbb{Z}$).

For any $a, b \in n\mathbb{Z}$, $a = kn$ and $b = mn$ for some integers $k, m$. Then $a-b = kn - mn = (k-m)n \in n\mathbb{Z}$. So, $n\mathbb{Z}$ is closed under subtraction.

For any $a, b \in n\mathbb{Z}$, $a = kn$ and $b = mn$. Then $a \cdot b = (kn)(mn) = (kmn)n \in n\mathbb{Z}$. So, $n\mathbb{Z}$ is closed under multiplication.

Thus, $n\mathbb{Z}$ is a subring of $\mathbb{Z}$. (True)
Statement II: A ring $\mathbb{Z}_m$ is an integral domain if and only if $m$ is a prime number. Here, $m=6$, which is not a prime number ($6 = 2 \times 3$). In $\mathbb{Z}_6$, we have $2 \neq 0$ and $3 \neq 0$, but $2 \times_6 3 = 6 \equiv 0 \pmod 6$. This means $2$ and $3$ are zero divisors. Therefore, $(\mathbb{Z}_6, +_6, \times_6)$ is not an integral domain. (False)
Statement I is correct but Statement II is incorrect."

type="MCQ" question="[Year: 2023] Which one of the following rings is an integral domain?" options=["$\mathbb{Z}_{100}$","$\mathbb{Z}_{102}$","$\mathbb{Z}_{113}$","$\mathbb{Z}_{153}$"] answer="$\mathbb{Z}_{113}$" hint="Recall the condition for $\mathbb{Z}_n$ to be an integral domain." solution="A ring $\mathbb{Z}_n$ is an integral domain if and only if $n$ is a prime number.

• $100 = 10 \times 10$, which is composite. So $\mathbb{Z}_{100}$ is not an integral domain.


• $102 = 2 \times 51$, which is composite. So $\mathbb{Z}_{102}$ is not an integral domain.


• $113$ is a prime number. So $\mathbb{Z}_{113}$ is an integral domain.


• $153 = 3 \times 51$, which is composite. So $\mathbb{Z}_{153}$ is not an integral domain.

Therefore, $\mathbb{Z}_{113}$ is the only integral domain among the given options."

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Quick Practice

Is the set of $2 \times 2$ matrices with integer entries, $M_2(\mathbb{Z})$, an integral domain?

What is the characteristic of the ring $\mathbb{Z}_4 \times \mathbb{Z}_6$?

Let $S = \{a+b\sqrt{3} \mid a,b \in \mathbb{Q}\}$. Is $S$ a field under standard addition and multiplication?

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Remember

> The distinction between a general ring, a commutative ring, a ring with unity, an integral domain, and a field lies in the increasing stringency of properties related to commutativity, existence of identities, and inverses/absence of zero divisors. Focus on these definitions and their implications for finite rings.

See full notes for detailed explanations and worked examples!

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Part 2: Ideals

Ideals are specific subrings fundamental to understanding ring structure and homomorphisms. They serve as kernels of ring homomorphisms and are used to construct factor rings, analogous to normal subgroups in group theory. This topic explores their definitions, properties, and classifications like prime and maximal ideals.

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Formula Sheet

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| Concept | Formula | When to Use |

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| A subset $I$ of a ring $R$ is a (two-sided) ideal if:
1. $(I, +)$ is a subgroup of $(R, +)$ (i.e., $I \neq \emptyset$ and $a, b \in I \implies a-b \in I$).
2. For all $r \in R$ and $a \in I$, $ra \in I$ and $ar \in I$ (absorption property). | To check if a subset is an ideal. | | 2 | Left Ideal | A subset $I$ of a ring $R$ is a left ideal if:
1. $(I, +)$ is a subgroup of $(R, +)$.
2. For all $r \in R$ and $a \in I$, $ra \in I$. | For specific algebraic structures where left/right multiplication differs. | | 3 | Right Ideal | A subset $I$ of a ring $R$ is a right ideal if:
1. $(I, +)$ is a subgroup of $(R, +)$.
2. For all $r \in R$ and $a \in I$, $ar \in I$. | For specific algebraic structures where left/right multiplication differs. | | 4 | Ideal Test (Simplified) | A non-empty subset $I \subseteq R$ is an ideal if:
1. $a, b \in I \implies a-b \in I$.
2. $r \in R, a \in I \implies ra \in I$ and $ar \in I$. | Quick verification that $I$ is an ideal. | | 5 | Trivial Ideals | $\{0\}$ and $R$ itself. | Always ideals in any ring $R$. | | 6 | Principal Ideal | $I = \langle a \rangle = \{ra \mid r \in R\}$ (if $R$ is commutative with unity).
Or for general rings: $I = \{r_1 a s_1 + \dots + r_k a s_k \mid r_i, s_i \in R, k \in \mathbb{N}\}$. | To generate an ideal from a single element. | | 7 | Sum of Ideals | $I+J = \{i+j \mid i \in I, j \in J\}$. $I+J$ is the smallest ideal containing $I \cup J$. | To combine two ideals. | | 8 | Product of Ideals | $IJ = \{\sum_{k=1}^n i_k j_k \mid i_k \in I, j_k \in J, n \in \mathbb{N}\}$. $IJ \subseteq I \cap J$. | To define products of ideals. | | 9 | Intersection of Ideals | $I \cap J = \{x \mid x \in I \text{ and } x \in J\}$. $I \cap J$ is an ideal. | To find common elements of ideals; always an ideal. | | 10 | Quotient Ring (Factor Ring) | $R/I = \{r+I \mid r \in R\}$, where $I$ is an ideal of $R$.
Addition: $(a+I) + (b+I) = (a+b)+I$.
Multiplication: $(a+I)(b+I) = ab+I$. | To construct new rings from existing ones. | | 11 | First Isomorphism Theorem | If $\phi: R \to S$ is a ring homomorphism, then $R/\operatorname{ker}(\phi) \cong \text{Im}(\phi)$. | To relate factor rings to homomorphic images. | | 12 | Second Isomorphism Theorem | Let $I$ be an ideal of $R$ and $S$ be a subring of $R$. Then $(S+I)/I \cong S/(S \cap I)$. | To relate factor rings involving sums and intersections. | | 13 | Third Isomorphism Theorem | Let $I$ and $J$ be ideals of $R$ with $I \subseteq J$. Then $(R/I)/(J/I) \cong R/J$. | To simplify nested factor rings. | | 14 | Prime Ideal | An ideal $P \neq R$ in a commutative ring $R$ with unity is prime if $ab \in P \implies a \in P \text{ or } b \in P$. | To test for primality of an ideal. Also, $R/P$ is an integral domain. | | 15 | Maximal Ideal | An ideal $M \neq R$ in a commutative ring $R$ with unity is maximal if for any ideal $J$ such that $M \subseteq J \subseteq R$, either $J=M$ or $J=R$. | To test for maximality of an ideal. Also, $R/M$ is a field. | | 16 | Kernel of Homomorphism | $\operatorname{ker}(\phi) = \{r \in R \mid \phi(r)=0_S\}$ for $\phi: R \to S$. | To find the ideal associated with a homomorphism. |

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Key Definitions

A non-empty subset $I$ of a ring $R$ is an ideal if it is a subgroup under addition and satisfies the absorption property: for all $r \in R$ and $a \in I$, $ra \in I$ and $ar \in I$.

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Must Remember

Ideal vs. Subring: Every ideal is a subring, but not every subring is an ideal (e.g., $\mathbb{Z}$ is a subring of $\mathbb{Q}$, but not an ideal).

Kernel of Homomorphism: The kernel of any ring homomorphism $\phi: R \to S$ is always an ideal of $R$.

Factor Ring Properties:

* $R/I$ is an Integral Domain $\iff I$ is a Prime Ideal.
* $R/I$ is a Field $\iff I$ is a Maximal Ideal.

Relationship between Prime and Maximal: In a commutative ring with unity, every maximal ideal is a prime ideal. The converse is not always true (e.g., $\{0\}$ is a prime ideal in $\mathbb{Z}$, but not maximal).

Ideal Containing Unity: If an ideal $I$ of a ring $R$ with unity contains the multiplicative identity $1$, then $I=R$.

Existence of Maximal Ideals: Every proper ideal in a commutative ring with unity is contained in a maximal ideal (Krull's Theorem/Zorn's Lemma).

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Common Mistakes

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Quick Practice

type="MCQ" question="Which of the following statements is FALSE for a commutative ring $R$ with unity?" options=["Every maximal ideal is a prime ideal.","If $I$ is an ideal of $R$ such that $R/I$ is a field, then $I$ is maximal.","The intersection of any two ideals of $R$ is an ideal of $R$.","The union of any two ideals of $R$ is an ideal of $R$."] answer="The union of any two ideals of $R$ is an ideal of $R." solution="1. Every maximal ideal is a prime ideal: TRUE. If$ R/M $is a field, it is an integral domain, so$ M $ is prime.

If $I$ is an ideal of $R$ such that $R/I$ is a field, then $I$ is maximal: TRUE, by definition of maximal ideal.

The intersection of any two ideals of $R$ is an ideal of $R$: TRUE.

The union of any two ideals of $R$ is an ideal of $R$: FALSE. For example, in $\mathbb{Z}$, $I=\langle 2 \rangle$ and $J=\langle 3 \rangle$ are ideals. $I \cup J = \{..., -4, -3, -2, 0, 2, 3, 4, ...\}$. $2 \in I \cup J$ and $3 \in I \cup J$, but $2+3=5 \notin I \cup J$. Thus, $I \cup J$ is not closed under addition and hence not an ideal."

type="MCQ" question="Let $R = \mathbb{Z}[x]$ be the ring of polynomials with integer coefficients. Which of the following is a prime ideal but NOT a maximal ideal?" options=["$\langle x \rangle$","$\langle 2 \rangle$","$\langle 2, x \rangle$","$\langle 0 \rangle$"] answer="$\langle x \rangle$" solution="1. $\langle x \rangle$: $R/\langle x \rangle \cong \mathbb{Z}$. $\mathbb{Z}$ is an integral domain (so $\langle x \rangle$ is prime), but not a field (so $\langle x \rangle$ is not maximal).

$\langle 2 \rangle$: $R/\langle 2 \rangle \cong \mathbb{Z}_2[x]$. $\mathbb{Z}_2[x]$ is an integral domain (so $\langle 2 \rangle$ is prime), but not a field (so $\langle 2 \rangle$ is not maximal).

$\langle 2, x \rangle$: $R/\langle 2, x \rangle \cong \mathbb{Z}_2$. $\mathbb{Z}_2$ is a field (so $\langle 2, x \rangle$ is maximal, and thus prime).

$\langle 0 \rangle$: $R/\langle 0 \rangle \cong \mathbb{Z}[x]$. $\mathbb{Z}[x]$ is an integral domain, so $\langle 0 \rangle$ is prime. However, $\mathbb{Z}[x]$ is not a field, and there are proper ideals containing $\langle 0 \rangle$ (e.g., $\langle x \rangle$), so it is not maximal.

Both $\langle x \rangle$ and $\langle 2 \rangle$ are prime but not maximal. The question asks for 'a' prime ideal but not maximal. Both $\langle x \rangle$ and $\langle 2 \rangle$ fit. Let's pick $\langle x \rangle$ as it is typically a more direct example for $\mathbb{Z}[x]$."

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Remember

> Ideals are the building blocks for constructing factor rings, and the properties of these factor rings (integral domain, field) directly reveal the nature (prime, maximal) of the ideal itself.

See full notes for detailed explanations and worked examples!

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Part 3: Prime and Maximal Ideals

Prime and maximal ideals are fundamental structures in ring theory, crucial for understanding the properties of quotient rings. They generalize the concepts of prime numbers and maximal subgroups, providing insight into when a quotient ring forms an integral domain or a field.

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Formula Sheet

tip title="Next Up"
Proceeding to Fields and Quotient Fields.

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Part 4: Fields and Quotient Fields

This section reviews fundamental concepts of fields, their construction, and properties, including integral domains, finite fields, and quotient fields. Focus is on essential definitions, theorems, and their applications in solving problems.

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Formula Sheet

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| Concept | Formula | When to Use |

| | 1 | Field Definition | A commutative ring $(F, +, \cdot)$ with unity $1 \neq 0$ where every non-zero element has a multiplicative inverse. | $\mathbb{Q}, \mathbb{R}, \mathbb{C}, \mathbb{Z}_p$ (for prime $p$) | | 2 | Integral Domain (ID) | A commutative ring $(D, +, \cdot)$ with unity $1 \neq 0$ and no zero divisors. | $\mathbb{Z}, \mathbb{Z}[x], \mathbb{Z}_p$ (for prime $p$) | | 3 | Relation: Finite ID is a Field | Every finite integral domain is a field. | $\mathbb{Z}_5$ is an ID and finite, so it is a field. | | 4 | Characteristic of a Field $F$ | $\text{char}(F) = n$, the smallest positive integer such that $n \cdot 1 = 0$. If no such $n$ exists, $\text{char}(F) = 0$. | $\text{char}(\mathbb{Z}_p) = p$, $\text{char}(\mathbb{Q}) = 0$. | | 5 | Prime Field | The smallest subfield of any field $F$. Isomorphic to $\mathbb{Q}$ (if $\text{char}(F)=0$) or $\mathbb{Z}_p$ (if $\text{char}(F)=p$). | The prime field of $\mathbb{R}$ is $\mathbb{Q}$. The prime field of $\mathbb{F}_{p^n}$ is $\mathbb{Z}_p$. | | 6 | Quotient Field Construction | For an integral domain $D$, its quotient field $Q(D)$ consists of equivalence classes of fractions $\frac{a}{b}$ where $a, b \in D, b \neq 0$. | $Q(\mathbb{Z}) = \mathbb{Q}$. $Q(\mathbb{Z}[x]) = \mathbb{Q}(x)$. | | 7 | Order of Finite Fields (Galois Fields) | The order of any finite field $\mathbb{F}$ is $p^n$ for some prime $p$ and integer $n \ge 1$. | $\mathbb{F}_4, \mathbb{F}_8, \mathbb{F}_9, \mathbb{F}_{27}$ are possible orders. $\mathbb{F}_6, \mathbb{F}_{10}$ are not. | | 8 | Uniqueness of Finite Fields | For every prime $p$ and integer $n \ge 1$, there exists a unique finite field of order $p^n$ (up to isomorphism), denoted $\mathbb{F}_{p^n}$ or $\text{GF}(p^n)$. | All fields of order $p^n$ are isomorphic. | | 9 | Subfields of Finite Fields | $\mathbb{F}_{p^n}$ contains a subfield of order $p^m$ if and only if $m$ divides $n$. This subfield is unique. | $\mathbb{F}_{2^{12}}$ has subfields of order $2^1, 2^2, 2^3, 2^4, 2^6, 2^{12}$ (divisors of 12 are 1, 2, 3, 4, 6, 12). | | 10 | Quotient Ring as a Field | For a commutative ring $R$ with unity, the quotient ring $R/I$ is a field if and only if $I$ is a maximal ideal of $R$. | $\mathbb{Z}/<p>$ is a field if $p$ is prime, because $<p>$ is a maximal ideal in $\mathbb{Z}$. |

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Key Definitions

If $K$ is a subfield of $F$, then $F$ is called an extension field of $K$. This is denoted as $F/K$. The degree of the extension is $[F:K] = \dim_K(F)$.

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Must Remember

Field $\implies$ Integral Domain. However, an integral domain is not necessarily a field (e.g., $\mathbb{Z}$).

Finite Integral Domain $\iff$ Field. This is a crucial distinction for finite rings.

Order of Finite Fields: Always of the form $p^n$ for a prime $p$ and integer $n \ge 1$.

Subfield Structure: $\mathbb{F}_{p^n}$ has a unique subfield of order $p^m$ for each divisor $m$ of $n$. The number of subfields is $\tau(n)$ (number of divisors of $n$).

$\mathbb{Z}_n$ as a Field: The ring $\mathbb{Z}_n$ (integers modulo $n$) is a field if and only if $n$ is a prime number. This is equivalent to saying that $<n>$ is a maximal ideal in $\mathbb{Z}$.

Characteristic: The characteristic of a finite field is always a prime number. The characteristic of an infinite field can be 0 or a prime number.

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Common Mistakes

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Previous Year Questions

type="MCQ" question="[Year: 2025] The integral domain of which cardinality is not possible:" options=["A and B only","A and C only","B and D only","C and D only"] answer="B and D only" hint="Recall that a finite integral domain is a field. What are the possible cardinalities of finite fields?" solution="A finite integral domain is always a field. The order (cardinality) of any finite field must be of the form $p^n$ for some prime $p$ and integer $n \ge 1$.
The given options are:
A. $5$: $5$ is a prime number ($5^1$). So $\mathbb{Z}_5$ is a field and thus an integral domain. Possible.
B. $6$: $6 = 2 \times 3$. Not of the form $p^n$. $\mathbb{Z}_6$ is not an integral domain (e.g., $2 \cdot 3 = 0 \pmod 6$). Not possible.
C. $7$: $7$ is a prime number ($7^1$). So $\mathbb{Z}_7$ is a field and thus an integral domain. Possible.
D. $10$: $10 = 2 \times 5$. Not of the form $p^n$. $\mathbb{Z}_{10}$ is not an integral domain (e.g., $2 \cdot 5 = 0 \pmod{10}$). Not possible.
Therefore, cardinalities 6 and 10 are not possible for an integral domain."

type="MCQ" question="[Year: 2022] Let $(\mathbb{F}, +, \cdot)$ be a finite field, then order of $\mathbb{F}$ can not be" options=["6","4","9","8"] answer="6" hint="The order of a finite field must be a power of a prime number." solution="The order of any finite field must be of the form $p^n$ for some prime number $p$ and integer $n \ge 1$.
Let's check the given options:

$6$: $6 = 2 \times 3$. This is not a power of a single prime. Thus, a field of order 6 does not exist.

$4$: $4 = 2^2$. This is a power of prime 2. A field of order 4 exists (e.g., $\mathbb{F}_4$).

$9$: $9 = 3^2$. This is a power of prime 3. A field of order 9 exists (e.g., $\mathbb{F}_9$).

$8$: $8 = 2^3$. This is a power of prime 2. A field of order 8 exists (e.g., $\mathbb{F}_8$).

Therefore, the order of $\mathbb{F}$ cannot be 6."

type="MCQ" question="[Year: 2024] Ler F be a field of order $16384$ then the number of proper subfields of F is :" options=["$6$","$3$","$4$","$8$"] answer="$3$" hint="The number of subfields of $\mathbb{F}_{p^n}$ is the number of divisors of $n$. Proper subfields exclude the field itself." solution="Let $F$ be a field of order $16384$.
First, express the order as a prime power $p^n$.
$16384 = 2^{14}$. So, $p=2$ and $n=14$.
A finite field $\mathbb{F}_{p^n}$ has a unique subfield of order $p^m$ for every positive divisor $m$ of $n$.
The number of subfields is equal to the number of divisors of $n$.
For $n=14$, the divisors are $1, 2, 7, 14$.
So, there are $\tau(14) = 4$ subfields. These subfields have orders $2^1, 2^2, 2^7, 2^{14}$.
Proper subfields are all subfields except the field $F$ itself (which has order $2^{14}$).
The proper subfields correspond to the divisors $1, 2, 7$.
Thus, there are 3 proper subfields."

type="MCQ" question="[Year: 2022] Let $(\mathbb{Z}, +, \bullet)$ be the ring of integers. Let $<m>$ for $m \in \mathbb{Z}$ be the ideal generated by $m$. Then which one is a field ?" options=["$\frac{\mathbb{Z}}{<3>}$","$\frac{\mathbb{Z}}{<4>}$","$\frac{\mathbb{Z}}{<8>}$","$\frac{\mathbb{Z}}{<6>}$"] answer="$\frac{\mathbb{Z}}{<3>}$" hint="A quotient ring $R/I$ is a field if and only if $I$ is a maximal ideal. For $\mathbb{Z}$, an ideal $<m>$ is maximal if and only if $m$ is a prime number." solution="For the ring of integers $\mathbb{Z}$, the quotient ring $\mathbb{Z}/<m>$ is a field if and only if the ideal $<m>$ is a maximal ideal. In $\mathbb{Z}$, an ideal $<m>$ is maximal if and only if $m$ is a prime number.
Let's check the given options:

$\frac{\mathbb{Z}}{<3>}$: Here $m=3$. Since $3$ is a prime number, $<3>$ is a maximal ideal in $\mathbb{Z}$. Therefore, $\mathbb{Z}/<3>$ is a field (specifically, it is $\mathbb{Z}_3$).

$\frac{\mathbb{Z}}{<4>}$: Here $m=4$. Since $4$ is not a prime number, $<4>$ is not a maximal ideal in $\mathbb{Z}$. For example, $<4> \subsetneq <2> \subsetneq \mathbb{Z}$. So, $\mathbb{Z}/<4>$ is not a field (it is $\mathbb{Z}_4$, which has zero divisors like $\bar{2} \cdot \bar{2} = \bar{0}$).

$\frac{\mathbb{Z}}{<8>}$: Here $m=8$. Since $8$ is not a prime number, $<8>$ is not a maximal ideal. So, $\mathbb{Z}/<8>$ is not a field.

$\frac{\mathbb{Z}}{<6>}$: Here $m=6$. Since $6$ is not a prime number, $<6>$ is not a maximal ideal. So, $\mathbb{Z}/<6>$ is not a field.

Thus, only $\mathbb{Z}/<3>$ is a field."

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Quick Practice

Is the ring $\mathbb{Z}[\sqrt{-5}] = \{a+b\sqrt{-5} \mid a,b \in \mathbb{Z}\}$ an integral domain? Is it a field?
* Solution: Yes, it is an integral domain because it is a subring of $\mathbb{C}$ (which is a field) and contains $1$. No, it is not a field, as elements like $2$ do not have multiplicative inverses in $\mathbb{Z}[\sqrt{-5}]$ (i.e., $1/2$ is not of the form $a+b\sqrt{-5}$ for $a,b \in \mathbb{Z}$).

What is the characteristic of the field $\mathbb{F}_{81}$?

* Solution: The order of $\mathbb{F}_{81}$ is $81 = 3^4$. The characteristic of a finite field $\mathbb{F}_{p^n}$ is always $p$. Therefore, $\text{char}(\mathbb{F}_{81}) = 3$.

How many distinct subfields does the field $\mathbb{F}_{2^{30}}$ have?

* Solution: The number of subfields of $\mathbb{F}_{p^n}$ is the number of positive divisors of $n$. Here, $n=30$. The divisors of $30$ are $1, 2, 3, 5, 6, 10, 15, 30$. There are $\tau(30) = 8$ distinct subfields.

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Remember

> A finite integral domain is always a field, and its order must be a prime power $p^n$.

See full notes for detailed explanations and worked examples!

Chapter Summary

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Chapter Review Questions

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What's Next?