My own notes about the MA249 Algebra II: Groups and Rings revision, mainly from the notes and example sheets.
- MA249-Algebra-II-Groups-and-Ring-Revision
- Lecture notes
- Example sheets
- Order of Groups, Orders of Elements
- Cyclic Groups
- $\mathbb{Z}_ {n}$ group
- Isomorphisms and Isomorphic groups
- Permutation Groups
- Dihedral Groups
- Subgroups
- Cosets and Lagrange's Theorem
- Normal Subgroups
- Direct Products and Groups of Order 4
- Generators
- Defining relations
- Homomorphisms and Quotient Groups
- Kernels and Images
- The Isomorphism Theorems
- Group Actions
- Orbits and Stabilisers
- Conjugation Action and Conjugacy Classes
- Conjugacy Classes in Alternating Groups
- Simple Groups
- Sylow's subgroup
- Rings and Subrings
- Isomorphisms and direct products
- Integral domains and fields
- Polynomials
- Ideals and Quotient Rings
- Quotient Rings
- Domains
- Prime and Irreducible Elements
- Unique Factorisation Domains
- Fields
- Polynomials
- Eisenstein's Criterion
- Fields of fractions
- Gauss's Lemma
First thing first, feel free to download the 2022-2023 lecture notes for MA249 Algebra II Groups and Rings.
MA249_Algebra_II_tutorial_sheet.pdf
- Let
$G$ be a group. The number of elements in$G$ is called the order of$G$ and is denoted by$|G|$ . This may be finite or infinite. - Let
$g\in G$ , then the order of$g$ , denoted by$|g|$ is the least integer$n > 0$ such that$g^{n} = 1$ , if such an$n$ exists. If there is no such$n$ , then$g$ has infinite order and we write$|g| = \infty$ .
-
$|g| = 1$ if and only if$g = 1$ . - If
$|g| = n$ , then for$k\in\mathbb{Z}$ ,$g^{k} = 1$ if and only if$n\vert k$ .
- A group
$G$ is called cyclic if it consists of the integral powers of a single element. i.e. for every$h\in G$ , there exists$k\in\mathbb{Z}$ with$g^{k} = h$ . The element$g$ is called a generator of$G$ . - In an infinite cyclic group, every generator
$g$ has infinite order. In a finite cyclic group of order$n$ , every generator$g$ has order$n$ . - All cyclic groups are abelian.
-
$\mathbb{Z}_ {n} = \set{0,1,2,...,n-1}$ , forming a group under the operation of addition modulo$n$ .
- An isomorphism
$\phi:G\to H$ between two groups$G$ and$H$ is a bijection from$G$ to$H$ such that$\phi(g_{1}g_{2}) = \phi(g_{1})\phi(g_{2})$ for all$g_{1},g_{2}\in G$ . - Two groups
$G$ and$H$ are called isomorphic if there is an isomorphism between them. In this case we write$G\cong H$ .
- If
$\phi:G\to H$ is an isomorphism, then$\phi(1_{G}) = 1_{H}$ and$\phi(g^{-1}) = \phi(g)^{-1}$ for all$g\in G$ . - If
$\phi:G\to H$ is an isomorphism, then$|g| = |\phi(g)|$ for all$g\in G$ .
If
-
$\phi(a_{i}) = a_{i+1}$ for$1\leq i< r$ . -
$\phi(a_{r}) = a_{1}$ and -
$\phi(b) = b$ for$b\in X\setminus\set{a_{1},a_{2},...,a_{r}}$ .
- A permutation is called even if it is a product of an even number of transpositions, and odd if it is a product of an odd number of transpositions.
- A cycle of even length is odd and a cycle of odd length is even. The product is seen as additivity when determining even or odd, e.g. when there are three 3-cycle with odd length, then each cycle is even and the product of it is even.
- The dihedral group of order
$2n$ consists of the isometries of$P$ , where$P$ is a regular$n$ -sided polygon in the plane. -
$n$ reflections of$P$ are the elements$a^{k}b$ for$0\leq k < n$ . Thus we have$$G = \set{a^{k}|0\leq k < n}\cup\set{a^{k}b|0\leq k < n}.$$
- A subset
$H$ of a group$G$ is called a subgroup of$G$ if it forms a group under the same operation as that of$G$ . - If
$H$ is a subgroup of$G$ , we write$H\leq G$ .
- If
$H$ is a subgroup of$G$ , then the identity element$1_{H}$ of$H$ is equal to the identity element$1_{G}$ of$G$ . - Let
$G$ be a group,$H\leq G$ and$K\leq G$ , then$H\cap K$ is itself a subgroup of$G$ .
Let
-
$h_{1},h_{2}\in H\implies h_{1}h_{2}\in H$ and -
$h\in H\implies h^{-1}\in H$ .
- Let
$g\in G$ , then the left coset$gH$ is the subset$\set{gh|h\in H}$ of$G$ . Similarly, the right coset$Hg$ is the subset$\set{hg|h\in H}$ of$G$ .
The following are equivalent for
-
$k\in gH$ ; -
$gH = kH$ ; -
$g^{-1}k\in H$ .
- Let
$G$ be a finite group and$H$ a subgroup of$G$ . Then the order of$H$ divides the order of$G$ . - Let
$G$ be a finite group. Then for any$g\in G$ , the order$|g|$ of$g$ divides the order$|G|$ of$G$ .
- The number of distinct left cosets of
$H$ in$G$ is called the index of$H$ in$G$ and it's written as$|G:H|$ . - Let
$G$ be a finite group and$H$ a subgroup of$G$ . Then$$|G| = |H|\cdot|G:H|.$$
- A subgroup
$H$ of a group$G$ is called normal in$G$ if$gH = Hg$ for all$g\in G$ .
- If
$G$ is any group and$H$ is a subgroup with$|G:H| = 2$ , then$H$ is a normal subgroup of$G$ . - Let
$H$ be a subgroup of the group$G$ . Then$H$ is normal in$G$ if and only if$ghg^{-1}\in H$ for all$g\in G$ and$h\in H$ .
- Let
$G$ and$H$ be two (multiplicative) groups. The direct product$G\times H$ of$G$ and$H$ ot be the set$\set{(g,h)\vert g\in G, h\in H}$ of ordered pairs of elements from$G$ and$H$ , with the obvious component-wise multiplication of elements$(g_{1}, h_{1})(g_{2}, h_{2}) = (g_{1}g_{2}, h_{1}h_{2})$ for$g_{1},g_{2}\in G$ and$h_{1}, h_{2}\in H$ .
- A group of order 4 is isomorphic either to a cyclic group
$C_{4}$ or to a Klein Four Group$C_{2}\times C_{2}$ .
- Let
$G$ be a group having prime order$p$ . Then$G$ is cyclic, that is,$G\cong C_{P}$ .
- Let
$G$ be a group of order 6. Then$G\cong C_{6}$ or$G\cong D_{3}$ .
- The elements
$\set{g_{1},g_{2},...,g_{r}}$ of a group$G$ are said to generate$G$ if every element of$G$ can be obtained by repeated multiplication of$g_{i}$ and their inverses. - A group is cyclic if and only if it can be generated by a single element.
- Let
$G$ be a group of order$2n$ generated by two elements$a$ and$b$ that satisfy the equations$a^{n} = 1, b^{2} = 1$ and$ba = a^{-1}b$ . Then$G\cong D_{n}$ .
- The equations
$\set{a^{n} = 1,b^{2} = 1, ba = a^{-1}b}$ are called defining relations for$D_{n}$ , which means roughly that$D_{n}$ is the largest group generated by two elements$a$ and$b$ that satisfy these equations.
- Let
$G$ be a group of order$mn$ generated by two elements$a$ and$b$ that satisfy the equations$a^{m} = 1, b^{n} = 1$ and$ba = ab$ . Then$G\cong C_{m}\times C_{n}$ .
- Let
$G$ be a group of order 8 generated by two elements$a$ and$b$ that satisfy the equations$a^{4} = 1, b^{2} = a^{2}$ and$ba = a^{-1}b$ . Then$G\cong Q_{8}$ . - Let
$G$ be a group of order 8. Then$G$ is isomorphic to one of$C_{8}, C_{4}\times C_{2}, C_{2}\times C_{2}\times C_{2}, D_{4}$ and$Q_{8}$ . -
$Q_{8}$ is known as the quaternion group, where we can define as the subgroup of$GL(2,\mathbb{C})$ :
- Let
$N$ be a normal subgroup of a group$G$ , and let$g,h\in G$ . Then the product of any element in the coset$gN$ with any element in the coset$hN$ is equal to an element in the coset$ghN$ . - If
$N$ is a normal subgroup of$G$ and$gN,hN$ are cosets of$N$ in$G$ , then$(gN)(hN) = ghN$ . - Let
$N$ be a normal subgroup of a group$G$ . Then the set$G/N$ of left cosets$gN$ of$N$ in$G$ forms a group under multiplication of sets. - The group
$G/N$ is called the quotient group of$G$ by$N$ . - If
$G$ is finite, then$$|G/N| = |G:N| = |G|/|N|.$$
- Let
$G$ and$H$ be groups. A homomorphism$\phi$ from$G$ to$H$ is a map$\phi:G\to H$ such that$\phi(g_{1}g_{2}) = \phi(g_{1})\phi(g_{2})$ for all$g_{1},g_{2}\in G$ . - An injective homomorphism is called a monomorphism, i.e. if
$\phi(g_{1}) = \phi(g_{2})\implies g_{1} = g_{2}.$ - A surjective homomorphism is called an epimorphism, i.e.
$\text{im}(\phi)\in H$ . - An isomorphism is a homomorphism
$\phi$ which is a bijection.
- Let
$\phi:G\to H$ be a homomorphism. Then the kernel$\ker(\phi)$ of$\phi$ is defined to be the set of elements of$G$ that map onto$1_{H}$ , i.e.$$\ker(\phi) = \set{g\in G:\phi(g) = 1_{H}}.$$ Note that$\ker(\phi)$ always contains$1_{G}$ .
- Let
$\phi:G\to H$ be a homomorphism. Then$\phi(1_{G}) = 1_{H}$ amd$\phi(g^{-1}) = \phi(g)^{-1}$ for all$g\in G$ . - Let
$\phi:G\to H$ be a homomorphism. Then$\phi$ is injective if and only if$\ker(\phi) = \set{1_{G}}.$ - Let
$\phi:G\to H$ be a homomorphism. Then$\ker(\phi)$ is a normal subgroup of$G$ . - Let
$N$ be a normal subgroup of a group$G$ . Then the map$\pi:G\to G/N$ defined by$\pi(g) = gN$ is a homomorphism with kernel$N$ . - Let
$\phi:G\to H$ be a homomorphism. Then$\text{im}(\phi)$ is a subgroup of$H$ .
- Let
$\phi:G\to H$ be a homomorphism with kernel$K$ . Then$G/K\cong\text{im}(\phi)$ . More precisely, there is an isomorphism$\overline{\phi}: G/K\to\text{im}(\phi)$ defined by$\overline{\phi}(gK) = \phi(g)$ for all$g\in G$ .
Let
-
$HK = KH$ is a subgroup of$G$ , -
$H\cap K$ is a normal subgroup of$H$ , and -
$H/(H\cap K)\cong HK/K$ .
Let
-
$K$ is a normal subgroup of$H$ . -
$H/K$ is a normal subgroup of$G/K$ . -
$(G/K)/(H/K)\cong G/H$ .
Let
-
$1_{G}\cdot x = x$ for all$x\in X$ . -
$(gh)\cdot x = g\cdot(h\cdot x)$ for all$g,h\in G, x\in X$ .
This is also defined as a left action, while a right action can be defined as a map
- The kernel of an action
$\cdot$ of$G$ on$X$ is defined to be the kernel$K = \ker(\phi)$ of the homomorphism$\phi:G\to\text{Sym}(X)$ . So$$K = \set{g\in G\vert g\cdot x = x,\forall x\in X}.$$ - The action is said to be faithful if
$K = \set{1}.$
- Every group
$G$ is isomorphic to a permutation group.
- Let
$\cdot$ be an action of$G$ act on$X$ . We define a relation$\sim$ on$X$ by$x\sim y$ if and only if there exists a$g\in G$ with$y = g\cdot x$ . The equivalence classes of$\sim$ are called the orbits of$G$ on$X$ . In particular, the orbit of a specific element$x\in X$ , which is denoted by$G\cdot x$ or by$\text{Orb}_ {G}(x)$ is$$G\cdot x = \text{Orb}_ {G}(x) = \set{y\in X:\text{there exists} g\in G\text{with}g\cdot x = y} = \set{g\cdot x:g\in G}.$$ - An action of
$G$ on$X$ is transitive if it has only a single orbit. Equivalently, an action is transitive if for every$x,y\in X$ , there is some$g\in G$ such that$g\cdot x = y$ .
- Let
$G$ act on$X$ and let$x\in X$ . Then the stabiliser of$x$ in$G$ , denoted by$G_{x}$ or$\text{Stab}_ {G}(x)$ , is$$\set{g\in G:g\cdot x = x}.$$ That is, the subset of$G$ comprising all elements that leave$x$ fixed. - The stabiliser is not just a subset of
$G$ , but actually a subgroup. - Let
$G$ act on$X$ and$x\in X$ . Then$\text{Stab}_ {G}(x)$ is a subgroup of$G$ and$\cap_{x\in X}\text{Stab}_ {G}(x)$ is the kernel of the action of$G$ on$X$ .
- Let a finite group
$G$ act on$X$ , and let$x\in X$ . Then$|G| = |\text{Orb}_ {G}(x)|\times |\text{Stab}_ {G}(x)|.$
- Another important action of
$G$ on$X = G$ , which is defined by$$g\cdot x = gxg^{-1}$$ for$g,x\in G$ . This action is called conjugation. - The orbits of the action are called the conjugacy classes of
$G$ , and elements in the same conjugacy class are said to be conjugate in$G$ . - So
$g,h\in G$ are conjugate if and only if there exists$f\in G$ with$h = fgf^{-1}$ . We will write$\text{Cl}_ {G}(g)$ for the orbit of$g$ , that is the conjugacy class containing$g$ . Thus,$$\text{Cl}_ {G}(g) = \set{xgx^{-1}\vert x\in G}.$$
- The centraliser of
$g$ in$G$ is written as$C_{G}(g)$ , that is,$$C_{G}(g) = \set{x\in G\vert gx = xg}.$$ - Let
$G$ be a finite group and let$g\in G$ . Then$|\text{Cl}_ {G}(g)| = |G|/|C_{G}(g)|.$
- The kernel
$K$ of the action consists of those$f\in G$ that fix and hence commute with all$g\in G$ . This is called the centre of$G$ and is denoted by$Z(G)$ . So we have$$Z(G) = \set{f\in G:fg = gf\quad\forall g\in G}.$$ - Note that
$g\in Z(G)$ if and only if$\text{Cl}_ {G}(g) = \set{g}.$
- Let
$G = S_{n}$ and$H = A_{n}$ . Let$h\in H$ , then$\text{Cl}_ {H}(h) = \text{Cl}_ {G}(h)$ or$|\text{Cl}_ {H}(h)| = \frac{1}{2}|\text{Cl}_ {G}(h)|.$
- A group
$G$ with$|G| > 1$ is called simple if its only normal subgroups are$G$ and$\set{1}$ . - A simple abelian group is cyclic of prime order.
- A subgroup
$H$ of a group$G$ is normal in$G$ if and only if$H$ consists of a union of conjugacy classes of$G$ . - The group
$A_{5}$ is simple.
- Let
$G$ be a finite group of order$p^{n}\cdot m$ , where$n$ is the largest power of the prime$p$ that divides$|G|$ , so$m$ is not divisible by$p$ . A subgroup of$G$ of order$p^{n}$ is called a Sylow p-subgroup of$G$ .
Let
-
$G$ has a Sylow p-subgroup, and any subgroup of$G$ of order$p^{a}$ for$1\leq a\leq n$ is contained in a Sylow p-subgroup of$G$ . - Any two Sylow p-subgroups of
$G$ are conjugate in$G$ . - The number
$r$ of Sylow p-subgroups of$G$ satisfies$r\equiv 1(\mod p)$ and$r\vert m$ .
- Sylow's second theorem says any two Sylow p-subgroups of
$G$ are conjugate in$G$ , and here conjugate means there is a homomorphism$\phi:G\to\text{Sym}(X)$ and$\text{Sym}(X)\cong S_{|X|}$ .
- Let
$G$ be a group of order$p^{n}m$ with$n\geq 1$ and$p \not\vert m$ . Let$$\text{Syl}_ {p}(G) = \set{H\leq G\vert|H| = p^{n}}$$ be the set of Sylow p-subgroups of$G$ . - If
$P\in\text{Syl}_ {p}(G)$ and$g\in G$ , then$gPg^{-1}\in\text{Syl}_ {p}(G)$ . -
$|\text{Syl}_ {p}(G)|$ divides$m = |G|/|P|$ . - If there is only one Sylow p-subgroup of
$G$ , then it is a normal subgroup of$G$ . - There are no simple groups of order 24.
A ring is a set
-
$(R,+)$ is an abelian group. -
$(ab)c = a(bc)$ for all$a,b,c\in R$ . -
$(a+b)c = ac+bc$ and$a(b+c) = ab+ac$ for all$a,b,c\in R$ . - There exists an element
$1 = 1_{R}\in R$ such that$1a = a1 = a$ for all$a\in R$ .
- A ring
$R$ is commutative if it satisfies $$ab = ba$$for all$a,b\in R$ . -
$\mathbb{Z},\mathbb{Q},\mathbb{R} and \mathbb{C}$ are all commutative rings with their usual addition and multiplication. -
$\mathbb{Z}_ {n}$ is a commutative ring under addition and multiplication modulo$n$ , for every positive integer$n$ .
- A subset
$S$ of a ring$R$ is called a subring of$R$ if it forms a ring under the same operations as$R$ with the same identity element.
Let
-
$S$ is a subgroup of$(R,+)$ . -
$a_{1},a_{2}\in S\implies a_{1}a_{2}\in S$ and -
$1_{R}\in S$ .
- The intersection of any set of subrings of
$R$ is itself a subring.
A map
-
$\phi$ is a bijection. -
$\phi(r_{1} + r_{2}) = \phi(r_{1})+\phi(r_{2})$ for all$r_{1},r_{2}\in R$ and -
$\phi(r_{1}r_{2}) = \phi(r_{1})\phi(r_{2})$ for all$r_{1},r_{2}\in R$ .
Two rings
Let
-
$\phi(0_{R}) = 0_{S}$ . -
$\phi(1_{R}) = 1_{S}$ .
Let
- The rings
$\mathbb{Z}_ {m}\times\mathbb{Z}_ {n}$ and$\mathbb{Z}_ {mn}$ are isomorphic if and only if$m$ and$n$ are coprime.
- If
$n = p_{1}^{a_{1}}...p_{k}^{a_{k}}$ is a decomposition of$n$ into a product of distinct primes then$$\mathbb{Z}_ {n}\cong \mathbb{Z}_ {p_{1}^{a_{1}}}\times...\times\mathbb{Z}_ {p_{k}^{a_{k}}}$$ as rings.
- If
$a$ and$b$ are non-zero elements of a ring$R$ with$ab = 0$ , then$a$ and$b$ are called zero divisors. - A ring
$R$ is called an integral domain (or just domain) if$R$ is commutative, nonzero and has no zero divisors, that is if$a,b\in R, ab = 0$ implies$a = 0$ or$b = 0$ . - The rings
$\mathbb{Z},\mathbb{Q},\mathbb{R}$ and$\mathbb{C}$ are all integral domains. -
$\mathbb{Z}_ {n}$ is a domain if and only if$n$ is prime.
- An element
$a$ of a ring$R$ is called a unit if it has a two-sided inverse under multiplication; that is, if there exists$b\in R$ with$ab = ba = 1$ .
- A non-zero ring
$R$ is called a division ring if$R\setminus\set{0}$ is a group under multiplication; that is, if all of its non-zero elements are units. - A field is a commutative division ring.
- Every field is an integral domain.
- A finite integral domain is a field.
- Let
$R$ be a ring. If there exists a positive integer$n$ such that$nx = 0$ for all$x\in R$ , then we call the smallest such positive integer is the characteristic of$R$ . If there is no such positive integer, then we say the charateristic of$R$ is 0. -
$\mathbb{Z}_ {n}$ has characteristic$n$ . -
$\mathbb{Z}$ and$\mathbb{Q}$ have characteristic 0. - The polynomial ring
$R[x]$ has the same characteristic as$R$ .
- If
$R$ is an integral domain, then so is$R[x]$ . - If
$R$ is an integral domain, then the units in$R$ and in$R[x]$ are the same.
- For any
$f,g\in F[x]$ with$0\ne g$ , there exists$q,r\in F[x]$ with$f = qg + r$ , where either$r = 0$ or$\deg(r) < \deg(g).$
- Let
$f = f(x)\in F[x]$ . Then for$a\in F$ ,$f(a) = 0$ if and only if$(x-a)$ divides$f$ .
Let
-
$\phi(r_{1}+r_{2}) = \phi(r_{1})+\phi(r_{2})$ for all$r_{1},r_{2}\in R$ , -
$\phi(r_{1}r_{2}) = \phi(r_{1})\phi(r_{2})$ for all$r_{1},r_{2}\in R$ and -
$\phi(1_{R}) = 1_{S}$ .
- The image
$\text{im}(\phi)$ of a ring homomorphism is just its image as a function:$$\text{im}(\phi) = \set{\phi(r)| r\in R}.$$ - The kernel
$\ker(\phi)$ of a ring homomorphism is defined to be its kernel as a homomorphism of additive groups. That is,$$\ker(\phi) = \set{r\in R|\phi(r) = 0_{S}}.$$ -
$\phi$ is injective if and only if$\ker(\phi) = \set{0}.$
A subset
-
$I$ is a subgroup of$(R,+)$ ; - For all
$x\in R$ and$y\in I$ , we have$xy\in I$ and$yx\in I$ .
- An ideal
$I$ of$R$ contains$1_{R}$ only when$I = R$ . - Let
$\phi:R\to S$ be a ring homomorphism. Then$\ker(\phi)$ is an ideal in$R$ . - If
$I$ and$J$ are ideals of$R$ , then so is$I+J = \set{i+j|i\in I,j\in J}.$
- When
$R$ is a commutative ring, the subset$$(a) = \set{ra|r\in R}$$ consists of all multiples of$a$ in$R$ , is an ideal of$R$ . It is known as the principal ideal generated by$a$ . - It is often written as
$aR$ or$Ra$ .
- The cosets of an ideal form a ring under addition in the quotient group and the multiplication
$(I+a)(I+b) = I+ab$ .
- Let
$\phi:R\to S$ be a ring homomorphism with kernel$I$ . Then$R/I\cong\text{im}(\phi)$ . More precisely, there is an isomorphism$\overline{\phi}: R/I\to\text{im}(\phi)$ defined by$\overline{\phi}(I+a) = \phi(a)$ for all$a\in R$ .
- A domain
$R$ is called a principal ideal domain if every ideal of$R$ is principal. - For every field
$F$ , the polynomial ring$F[x]$ is a principal ideal domain.
The following statements are equivalent for
-
$x\vert y$ ; -
$y\in (x)$ ; -
$(x)\supseteq (y)$ .
Let
-
$r$ is not a unit, - if
$r = ab$ with$a,b\in R$ , then either$a$ or$b$ is a unit.
We say that
-
$r$ is not a unit, - if
$r\vert xy$ , then$r\vert x$ or$r\vert y$ .
- If
$R$ is a domain, then every prime element of$R$ is irreducible. - If
$R$ is a principal ideal domain, then every irreducible element of$R$ is prime.
- An integral domain
$R$ is a factorisation domain (FD) if each non-unit$x\in R\setminus\set{0}$ admits a factorisation$x = r_{1}r_{2}...r_{n}$ where the$r_{i}$ are irreducible elements.
An factorisation domain
- it is a factorisation domain; and
- for each non-unit
$x$ in$R\setminus\set{0}$ and any two factorisations$x = r_{1}r_{2}...r_{n} = s_{1}s_{2}...s_{m}$ where all$r_{i}$ and$s_{i}$ are irreducible, we have$m = n$ , and there exists$\sigma\in S_{n}$ such that$r_{i}\sim s_{\sigma(i)}$ for all$i$ .
- If
$R$ is a UFD, then every irreducible element of$R$ is prime. - A principal ideal domain is a factorisation domain.
- If
$R$ is a fatorisation domain in which all irreducibles are primes, then$R$ is a unique factorisation domain. In particular, every principal ideal domain is a unique factorisation domain.
- An ideal
$I$ of a ring$R$ is called maximal, if$I\ne R$ , but if$J$ is any ideal of$R$ with$I\subseteq J\subseteq R$ , then$I = J$ or$J = R$ .
- An ideal
$I$ in a commutative ring$R$ is maximal if and only if$R/I$ is a field. - For any
$a\ne 0$ , the ideal$(a)$ in a principal ideal domain$R$ is maximal if and only if$a$ is irreducible.
- An element
$\alpha\in\mathbb{C}$ is said to be algebraic (over$\mathbb{Q}$ ) if it satisfies a polynomial equation$f(\alpha) = 0$ for some$f\in\mathbb{Q}[x]$ with$\deg(f) > 0$ . Otherwise$\alpha$ is called transcendental.
- If
$\alpha$ is an algebraic element of$\mathbb{C}$ , then there is a unique non-zero polynomial$m\in\mathbb{Q}[x]$ with leading coefficient 1 such that$m(\alpha) = 0$ and$m$ is irreducible. - By the First Isomorphism Theorem for Rings, we have
$$\text{im}(\phi_{\alpha})\cong\mathbb{Q}[x]/(f).$$ Since$f$ is irreducible,$(f)$ is a maximal ideal and hence$\mathbb{Q}[x]/(f)$ is a field. So$\text{im}(\phi_{\alpha})$ is a subfield of$\mathbb{C}$ , denoted by$\mathbb{Q}[x]$ . Fields of this type are called number fields.
- A field
$F$ is algebraically closed if for every$f(x)\in F[x]$ of degree at least 1, there exists$a\in F$ such that$f(a) = 0$ .
- An element
$0\ne f = a_{0} + a_{1}x + ... + a_{n}x^{n}\in R[x]$ is called primitive if$\gcd(a_{0},a_{1},...,a_{n}) = 1$ .
- Let
$R$ be a unique factorisation domain. Let$f = a_{0} + a_{1}x + ... + a_{n}x^{n}$ be a primitive polynomial in$R[x]$ , and suppose there is prime$p\in R$ such that$p\not\vert a_{n}$ and$p\vert a_{i}$ for$0\leq i < n$ and$p^{2}\not\vert a_{0}$ , Then$f$ is irreducible in$R[x]$ .
- We define an equivalence relation on
$W$ by$(a,b)\sim (c,d)$ whenever$ad = bc$ , where$$W = R\times(R\setminus\set{0}) = \set{(x,y)\in R\times R | y\ne 0}.$$ - An equivalence class of
$(a,b)$ is called a fraction and denoted$a/b$ .
- If
$R$ is a domain then$Q(R)$ is a field under the operations$$\frac{a}{b}+\frac{c}{d} = \frac{ad+bc}{bd},\quad \frac{a}{b}\cdot\frac{c}{d} = \frac{ac}{bd},$$ and$\pi:R\to Q(R)$ , with$\pi(r) = r/1$ is an injective ring homomorphism.
-
$Q = Q(R)$ is called the field of fractions of domain$R$ .
- The product of two primitive polynomials is primitive.
- Let
$R$ be a unique factorisation domain with field of fractions$Q = Q(R)$ . Then a primitive polynomial in$R[x]$ is irreducible if and only if it is irreducible in$Q[x]$ .
- A primitive irreducible polynomial in
$\mathbb{Z}[x]$ remains irreducible in$\mathbb{Q}[x]$ .
- If
$R$ is a unique fatorisation domain, then there are two kinds of irreducibles in$R[X]$ : irreducible elements in$R$ and primitive elements in$R[X]$ that are irreducible in$Q[X]$ . - If
$R$ is a unique fatorisation domain, then so is$R[x]$ .