Defn's/Statements

Question 1

Define a Noetherian Ring

Answer 1

R is noetherian, if every ideal of R is finitely generated

Question 2

What is a Euclidean Ring?

Answer 2

A ring equipped with a Euclidean Norm

Question 3

What is Hilberts Basis Theorem?

Answer 3

If K is noetherian therefore K[X] is a noetherian ring

Question 4

What is a Euclidean Norm?

Answer 4

A euclidean norm on a ring R is a function A : R → N such that for all a, b ∈ R{0}, there exists q ∈ R such that a = qb or N(a − qb)

Question 5

What is meant by saying that b is an associate of a?

Answer 5

b = xa where x is a unit of R

Question 6

What is meant by saying that a is irreducible in R?

Answer 6

a is not 0, not a unit and not a product of two non-units

Question 7

State Dickson’s Lemma about minimal monomials of S.

Answer 7

Let S be a subset of (M(X1, . . . ,Xn), |). Then:

1. Smin is finite.
2. Every element of S is divisible by at least one element of Smin.

Question 8

billy. Define a Prime in a domain

Answer 8

a non-unit p is called a prime if p/ab ⇒ p/a or p/b

Question 9

Define a monomial ordering and it’s conditions.

Answer 9

A monomial ordering is a partial order ≤ on M = M(X1, . . . ,Xn) which satisfies:

1) ≤ is a total order i.e. m, m’ ∈ M implies m ≤ m’ or m > m’
2) m ∈ M implies 1 ≤ m
3) if m’ ≤ m then for every m1 ∈ M , m1m’ ≤ m1m

Question 10

What is meant by a prime in a commutative domain R?

Answer 10

p ∈ R is a prime if p is not a unit and ∀a, b ∈ R, p∤a, p∤b ⇒ p∤ab.

Question 11

What is a unique factorisation domain (UFD)?

Answer 11

A domain where every non-unit is a product of primes

Question 12

What is meant by saying that an ideal I of a commutative ring R is a maximal ideal?

Answer 12

I ≠ R and there exists no such J: I ⊊ J ⊊ R

Question 13

What is meant by the radical, √I, of an ideal I?

Answer 13

√I = {a ∈ R | ∃n ∈ N : a^n ∈ I} √I is an ideal of R containing I

Question 14

What is a radical ideal?

Answer 14

I is a radical ideal if √I = I

Question 15

What is meant by the variety V(I) of an ideal I of K[X1, . . . , Xn]?

Answer 15

V(I) = {(a1, . . . , an) ∈ K^n | f(a1, . . . , an) = 0 ∀f ∈ I};

Question 16

What is meant by the ideal ℑ(S) of a set S ⊆ K^n?

Answer 16

I(S) = {f ∈ K[X1, . . . , Xn] | f(a1, . . . , an) = 0 ∀(a1, . . . , an) ∈ S}

Question 17

State the weak Nullstellensatz

Answer 17

If K is algebraically closed and I is a proper ideal of K[X1, . . ,Xn], then V(I) ≠ ∅.

Question 18

If R is a UFD, what can this tell us

Answer 18

R[X] and R[X,Y,..] are also UFDs

Question 19

Explain what is meant by saying that R satisfies the ascending chain condition (acc) on ideals.

Answer 19

Every ascending chain I1 ⊆ I2 ⊆ . . . of ideals of R eventually stabilises, that is, there exists n0: In0 = In for all n ≥ n0.

Question 20

If D is a commutative domain, define the notion of an irreducible element of D

Answer 20

r ∈ D is irreducible if r is not a unit, and in every factorisation r = st with s, t ∈ D, one of
s, t is a unit.

Question 21

Describe, without proof, all the irreducible elements of C[X].

Answer 21

{aX + b : a, b ∈ C, a does not equal 0} (or: polynomials of degree 1).

Question 22

Give the definition of a Groebner basis with respect to monomial ordering of an ideal of R

Answer 22

A Groebner basis of an ideal I is a finite subset {f1, f2, . . . , fm} of I \ {0} such lm≼( fi ) generate the ideal LT(I) of leading terms of I.

Question 23

What is √(√I) ?

Answer 23

√I

Question 24

What is meant by saying that I is a prime ideal?

Answer 24

I is a proper ideal of R such that a, b ∈ R\I ⇒ ab∈R\I
ie
Prime Ideal = {ar | a∈I, r∈R/I}

Question 25

What is a monomial ideal of K[X1, . . . , Xn]? Explain why the ideal {0} is a monomial ideal.

Answer 25

A monomial ideal of K[X1, . . . , Xn] is an ideal generated by a set of monomials in X1, . . . , Xn. The ideal {0} is a monomial ideal because it is generated by the empty set.

Question 26

Define LT≼( I ), the ideal of leading terms of I.

Answer 26

LT≼( I ) is the ideal generated by the leading monomials of the non-zero polynomials in I.

Question 27

What is a subring of a domain?

Answer 27

A domain

Question 28

What is a reduced Groebner basis?

Answer 28

A reduced Gr¨obner basis is a Gr¨obner basis {g1, . . . , gn} such that all the gi are monic polynomials and for any i ≠ j, the polynomial gi is reduced with respect to gj .

Question 29

What is meant by saying that a commutative ring R is a principal ideal domain?

Answer 29

R is a domain and every ideal of R can be generated by one element.

Question 30

Define the notion of a nilpotent element of R.

Answer 30

An element a ∈ R is nilpotent if there is n ∈ N such that a^n = 0.

Question 31

What is it to say that a field K is algebraically closed?

Answer 31

We say that a field K is algebraically closed if every non-constant polynomial in K[X] has a root in K

Question 32

State without proof Hilbert’s Basis Theorem for polynomials over a field.

Answer 32

If K is a field then K{X1,...Xn] is a noetherian ring

Question 33

A Euclidean ring is also another type of ring, what is this ring?

Answer 33

A principal ideal ring

Question 34

A principal ideal domain is what?

Answer 34

A domain is a domain whose ideals are generated by one element ( i.e. a PIR which is also a domain)

Question 35

A principal ideal domain (PIR which is a domain) is also another type of domain, what is this domain?

Answer 35

A unique factorisation domain (UFD)

Question 36

if K is a field, K[X] is a what type of domain?

Answer 36

A unique factorisation domain (UFD)

Question 37

We say that a polynomial f = an*X^n + . . . + a1*X + a0 ∈ R[X] is Eisenstein, if there is a prime p ∈ R such that ...?

Answer 37

1) p ∤ an 2) p | an-1, an-2, ....., a1, a0 3) p^2 ∤ a0

Question 38

What is the The Eisenstein Criterion?

Answer 38

If R is a UFD and f ∈ R[X] is Eisenstein, then f is irreducible in (ℱr R)[X].

Question 39

what is the nilradical Nil(R) of a ring R?

Answer 39

Nil(R) = √

Question 40

Every field R is a type of domain, what is this domain?

Answer 40

A unique factorisation domain (UFD)

Question 41

What is meant by saying that a polynomial f ∈ Q[X, Y, Z] is reduced modulo a set F ⊂ Q[X, Y, Z] \ {0} with respect to a monomial ordering ≼ ?

Answer 41

No monomial in f is divisible by the ≼-leading term of any of the elements of F

Question 42

Describe, without proof, all irreducible polynomials in R[X]

Answer 42

aX + b with a not equal to 0 and aX^2 + bX + c with b^2 − 4ac

Question 43

irreducible polynomials in Q[X] when degree ≥ 2?

Answer 43

are polynomials with no root in Q. to find this out, multiply up to get integer coefficents, and roots are of the form p/q where p | a_0 and q | a_n

Question 44

In a UFD, what's the relationship with primes and irreducible elements

Answer 44

Irreducible elements implies prime | Prime element implies irreducible, except 0.

Question 45

What is meant by a remainder of a polynomial f ∈ Q[X, Y, Z] modulo a set F ⊂ Q[X, Y, Z] with respect to a monomial ordering ≼ ?

Answer 45

A polynomial r ∈ Q[X, Y, Z] such that f can be reduced to r modulo F with respect to ≼ , and r is reduced modulo F.

Question 46

unit x unit = ? unit x non unit = ?

Answer 46

unit x unit = unit, unit x non unit = non unit

Question 47

solution of the membership problem for monomial ideals

Answer 47

let T⊂M[X] be a set of monomials. A polynomial f belongs to the ideal of K[X] every monomial in f is divisible by at least one monomial from T.

Question 48

if f mod G has a remainder, then..?

Answer 48

r is unique, as G is a groebner basis

Question 49

list the primes in ℝ

Answer 49

0 mate

Question 50

list the primes in ℝ[X]

Answer 50

since all primes are irreducible elements. | aX + b with a not equal to 0 and aX^2 + bX + c with b^2 − 4ac

Question 51

relationship between irreducibles and primes?

Answer 51

all primes are irreducible | in a UFD all irreducibles are also primes (except 0)

Question 52

what is 1+1

Answer 52

2

Question 53

subring test?

Answer 53

S is subset of R closed unider + closed under x identity ∈ S

Question 54

if F = ℱr R then R is a ?

Answer 54

subring of F

Question 55

what does having an eisenstain prime mean?

Answer 55

f is irreducible

Question 56

example of a UFD that is not principle?

Answer 56

K[X,Y] . as

Question 57

any examples if domains that arn't UFDs?

Answer 57

Z[√-5] | as 2x3 = (1 - √-5)(1 + √-5). 2 is irreducible in Z[√-5] but not prime

Question 58

example of a noetherian domain that is not a PID?

Answer 58

K[X,X,...Xn] n>1

Question 59

noetherian ring that is not a domain?

Answer 59

Zmod4 (or any mod, where the p is not prime)

Question 60

a domain that is not noetherian?

Answer 60

Q[X,....Xn,.....]

Question 61

R = domain | a maximal ideal is a ... ?

Answer 61

prime ideal