GATE Mathematics Algebra Flashcards

24 sample cards from the Algebra deck

Sampled from the end of the deck, so these are different cards from the ones shown on the syllabus page.

1. 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.

2. 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 $\mathbb{Z}$.

3. 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 $p$.

4. 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 $0 + I = I$.

5. 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 $R/\ker\varphi \cong \operatorname{im}\varphi$.

6. 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).

7. 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.

8. 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.

9. 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 $a$.

10. 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.

11. 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.

12. 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 $d(r) < d(b)$.

13. 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 $d(a+bi) = a^{2}+b^{2}$.

14. 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: $\deg(\text{remainder}) < \deg(\text{divisor})$.

15. 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.

16. 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 $\mathbb{Q}$.

17. 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 $\mathbb{Q}$.

18. 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 $\mathbb{F}_p$.

19. 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 $x \in \mathbb{F}_q$.

20. 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.

21. 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 $[K:F] = [K:L]\,[L:F]$.

22. 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.

23. 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.

24. 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. $\overline{\mathbb{Q}}/\mathbb{Q}$).

What this deck covers

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