GATE Mathematics Algebra Syllabus

Define a (left) group action of $G$ on a set $X$.

A map $G \times X \to X$, $(g,x) \mapsto g\cdot x$, satisfying $e \cdot x = x$ and $(gh)\cdot x = g\cdot(h\cdot x)$ for all $g,h \in G$, $x \in X$. Equivalently a homomorphism $G \to \operatorname{Sym}(X)$.

State the Orbit-Stabilizer Theorem.

For a group action of $G$ on $X$ and $x \in X$, $|\operatorname{Orb}(x)| = [G : \operatorname{Stab}(x)] = \dfrac{|G|}{|\operatorname{Stab}(x)|}$ when $G$ is finite, where $\operatorname{Stab}(x) = \{g : g\cdot x = x\}$.

State Burnside's (Cauchy-Frobenius) Lemma for counting orbits.

The number of orbits of a finite group $G$ acting on $X$ is $\dfrac{1}{|G|}\sum_{g \in G} |X^{g}|$, where $X^{g} = \{x : g\cdot x = x\}$ is the fixed-point set of $g$.

State the class equation of a finite group $G$.

$|G| = |Z(G)| + \sum_{i} [G : C_G(x_i)]$, where the sum runs over representatives $x_i$ of the conjugacy classes of size $>1$ and $C_G(x_i)$ is the centralizer.

Using the class equation, what can be said about the center of a $p$-group?

Any nontrivial finite $p$-group (order $p^{n}$, $n \geq 1$) has nontrivial center, $|Z(G)| > 1$. Each non-central class has size divisible by $p$, forcing $p \mid |Z(G)|$.

State Cauchy's Theorem.

If a prime $p$ divides $|G|$, then $G$ contains an element of order $p$ (equivalently a subgroup of order $p$).

For $|G| = p^{a}m$ with $p \nmid m$, what does Sylow's First Theorem guarantee?

$G$ has a Sylow $p$-subgroup: a subgroup of order $p^{a}$ (the full prime power). Moreover $G$ has subgroups of order $p^{k}$ for every $0 \le k \le a$.

State Sylow's Second and Third Theorems (conjugacy and number $n_p$).

Second: all Sylow $p$-subgroups are conjugate. Third: the number $n_p$ of Sylow $p$-subgroups satisfies $n_p \equiv 1 \pmod{p}$ and $n_p \mid m$ (where $|G| = p^{a}m$). Also $n_p = [G : N_G(P)]$.

When is a Sylow $p$-subgroup normal, in terms of $n_p$?

A Sylow $p$-subgroup $P$ is normal in $G$ iff $n_p = 1$ (it is the unique Sylow $p$-subgroup). Then $P \trianglelefteq G$.

Show how Sylow theory proves a group of order $15$ is cyclic.

$|G| = 15 = 3 \cdot 5$. $n_3 \mid 5$, $n_3 \equiv 1 \pmod 3 \Rightarrow n_3 = 1$; $n_5 \mid 3$, $n_5 \equiv 1 \pmod 5 \Rightarrow n_5 = 1$. Both Sylow subgroups normal, coprime orders $\Rightarrow G \cong \mathbb{Z}/3 \times \mathbb{Z}/5 \cong \mathbb{Z}/15$, cyclic.

Define a (two-sided) ideal $I$ of a ring $R$.

$I$ is an additive subgroup of $R$ such that for all $r \in R$ and $a \in I$, both $ra \in I$ and $ar \in I$ (absorption). It is the ring analogue of a normal subgroup.

In a commutative ring with unity, what are the only ideals of a field?

A field $F$ has exactly two ideals: the zero ideal $\{0\}$ and $F$ itself. Conversely, a commutative ring with $1 \neq 0$ whose only ideals are $\{0\}$ and itself is a field.

Define a prime ideal $P$ in a commutative ring $R$.

A proper ideal $P \subsetneq R$ such that whenever $ab \in P$, then $a \in P$ or $b \in P$. Equivalently, $R/P$ is an integral domain.

Define a maximal ideal $M$ and characterize it via the quotient.

A proper ideal $M \subsetneq R$ with no ideal strictly between $M$ and $R$. In a commutative ring with unity, $M$ is maximal iff $R/M$ is a field.

State the relationship between maximal and prime ideals.

In a commutative ring with unity, every maximal ideal is prime (since fields are integral domains), but not every prime ideal is maximal. E.g. $(0)$ is prime but not maximal in $\mathbb{Z}$.

In $\mathbb{Z}$, classify the prime and maximal ideals.

Every ideal is $(n)$. The ideal $(p)$ is maximal (and prime) iff $p$ is prime. $(0)$ is prime but not maximal. So nonzero prime ideals = maximal ideals = $(p)$ for primes $p$.

What is the quotient ring $R/I$ and how is its arithmetic defined?

$R/I$ is the set of cosets $r + I$ with $(a+I)+(b+I) = (a+b)+I$ and $(a+I)(b+I) = ab + I$, well-defined because $I$ is an ideal. Its zero element is $0 + I = I$.

State the First Isomorphism Theorem for rings.

If $\varphi: R \to S$ is a ring homomorphism, then $\ker\varphi$ is an ideal of $R$ and $R/\ker\varphi \cong \operatorname{im}\varphi$.

Define a Unique Factorization Domain (UFD).

An integral domain in which every nonzero non-unit factors as a product of irreducibles, and this factorization is unique up to order and unit multiples (associates).

In a UFD, what is the relationship between irreducible and prime elements?

In any integral domain every prime is irreducible; in a UFD the converse also holds, so irreducible $\iff$ prime. This equivalence characterizes much of UFD behavior.

State Gauss's Lemma / the key UFD fact about polynomial rings.

If $R$ is a UFD then the polynomial ring $R[x]$ is also a UFD. In particular $\mathbb{Z}[x]$ is a UFD, and the product of primitive polynomials is primitive.

Define a Principal Ideal Domain (PID).

An integral domain in which every ideal is principal, i.e. of the form $(a) = aR$ for some single element $a$.

State the chain of implications relating Euclidean domains, PIDs, and UFDs.

$\text{Field} \Rightarrow \text{Euclidean Domain} \Rightarrow \text{PID} \Rightarrow \text{UFD} \Rightarrow \text{Integral Domain}$. None of the reverse implications hold in general.

Give an example of a UFD that is not a PID, and a PID that is not Euclidean.

$\mathbb{Z}[x]$ is a UFD but not a PID (the ideal $(2,x)$ is not principal). $\mathbb{Z}\!\left[\frac{1+\sqrt{-19}}{2}\right]$ is a PID that is not a Euclidean domain.

Define a Euclidean Domain.

An integral domain $R$ with a function $d: R\setminus\{0\} \to \mathbb{Z}_{\geq 0}$ such that for all $a,b \in R$ with $b \neq 0$, there exist $q,r$ with $a = bq + r$ where $r = 0$ or $d(r) < d(b)$.

Give two standard examples of Euclidean domains and their norm functions.

$\mathbb{Z}$ with $d(n) = |n|$; a field's polynomial ring $F[x]$ with $d(f) = \deg f$; and the Gaussian integers $\mathbb{Z}[i]$ with $d(a+bi) = a^{2}+b^{2}$.

State the division behavior and units of the polynomial ring $F[x]$ over a field $F$.

$F[x]$ is a Euclidean domain (hence PID and UFD); the units are exactly the nonzero constants $F^{\times}$. Division algorithm: $\deg(\text{remainder}) < \deg(\text{divisor})$.

State the relationship between $a$ being a root of $f(x)$ and a linear factor.

For $f \in F[x]$ and $a \in F$: $f(a) = 0$ iff $(x - a) \mid f(x)$ (Factor Theorem). A degree-$n$ polynomial over a field has at most $n$ roots.

State Eisenstein's Irreducibility Criterion.

For $f(x) = a_n x^{n} + \cdots + a_0 \in \mathbb{Z}[x]$, if a prime $p$ satisfies $p \nmid a_n$, $p \mid a_i$ for all $i < n$, and $p^{2} \nmid a_0$, then $f$ is irreducible over $\mathbb{Q}$.

Use Eisenstein to show the $p$-th cyclotomic polynomial is irreducible.

$\Phi_p(x) = \frac{x^{p}-1}{x-1} = x^{p-1}+\cdots+1$. Substituting $x \to x+1$ gives a polynomial satisfying Eisenstein at $p$ (all middle coefficients $\binom{p}{k}$ divisible by $p$, constant term $p$ not divisible by $p^{2}$). Hence $\Phi_p$ is irreducible over $\mathbb{Q}$.

For a prime $p$ and integer $n \geq 1$, what is the order and uniqueness of a finite field $\mathbb{F}_q$?

A finite field has order $q = p^{n}$ for some prime $p$ and $n \geq 1$. For each such $q$ there exists a unique (up to isomorphism) field $\mathbb{F}_q$, the splitting field of $x^{q} - x$ over $\mathbb{F}_p$.

Describe the multiplicative group of a finite field $\mathbb{F}_q$.

$\mathbb{F}_q^{\times}$ is cyclic of order $q - 1$. A generator is called a primitive element, and every nonzero element satisfies $x^{q-1} = 1$, so $x^{q} = x$ for all $x \in \mathbb{F}_q$.

What is the characteristic of $\mathbb{F}_{p^n}$, and when is $\mathbb{F}_{p^m}$ a subfield of $\mathbb{F}_{p^n}$?

The characteristic is $p$. $\mathbb{F}_{p^{m}}$ is a subfield of $\mathbb{F}_{p^{n}}$ iff $m \mid n$. The Frobenius map $x \mapsto x^{p}$ is a field automorphism.

Define the degree $[K:F]$ of a field extension and state the tower (multiplicativity) law.

$[K:F]$ is the dimension of $K$ as a vector space over $F$. Tower law: if $F \subseteq L \subseteq K$ then $[K:F] = [K:L]\,[L:F]$.

What is the degree of $F(\alpha)/F$ when $\alpha$ is algebraic over $F$?

$[F(\alpha):F] = \deg m_{\alpha}(x)$, the degree of the minimal polynomial of $\alpha$ over $F$. A basis is $\{1, \alpha, \dots, \alpha^{d-1}\}$ where $d$ is this degree.

Define the minimal polynomial of an algebraic element $\alpha$ over $F$ and give its two key properties.

The unique monic polynomial $m_\alpha \in F[x]$ of least degree with $m_\alpha(\alpha)=0$. It is irreducible over $F$, and it divides every polynomial in $F[x]$ having $\alpha$ as a root.

What does it mean for an extension to be algebraic, and what is the relation to finite extensions?

$K/F$ is algebraic if every element of $K$ is a root of some nonzero polynomial in $F[x]$. Every finite extension is algebraic, but algebraic extensions can be infinite (e.g. $\overline{\mathbb{Q}}/\mathbb{Q}$).