CH 1 - Rings, Fields, and Polynomials Flashcards
Define a commutative ring with a 1.
Define an ideal, and hence state the 4 conditions an ideal satisfies.
Construct the additive cosets of I in a ring R.
When are 2 cosets equal? Can we form a group of cosets?
Define multiplication on the set of cosets. Can we form a ring of cosets? What is this ring called?
State a theorem on the quotient ring R/I.
Define a (ring) homomorphism.
Define the kernel of phi.
Define the image of phi.
State the First Isomorphism Theorem.
Define a zero divisor.
Define an integral domain.
Define a field.
What does F^x denote?
What is the implication between fields and integral domains?
Explain the notation in the photo.
Define the characteristic of F.
State the 3 conditions for K to be a subfield of F.
Proposition 1.11. (1) Every field is an integral domain.
(2) The set of non-zero elements in a field forms an abelian group under multiplication, denoted F^x.
State theorem where:
1 - F has characteristic zero
2 - F has characteristic p>0.
Define a proper ideal.
Define a maximal proper ideal.
State the Theorem on maximal J and quotient R/J.
Integral domains, inverses, fractions…
Describe the entire process of constructing a fraction field.
Define a polynomial over F.
Define F[X] and construct it as a commutative ring with a 1/ integral domain.
These Euclidean properties of F[X] allow us to divide polynomials usuiong the long-divison algorithm.
Also, every polynomial over F can be factorised as a product of irreducible polynomials uniquely (up to multiplication by scalars and reordering of factors).
Define an irreducible polynomial over F.
State a lemma on irreducible f and an ideal generated over f.
What are three techniques for determining whether a given polynomial is irreducible?
State them.
Gauss’ lemma, Eisenstein’s criterion and Reduction modulo prime.
(Gauss’ Lemma). Let f be a polynomial with integer coefficients that is
reducible over Q. Then it is reducible over Z as well. Furthermore, the degree of the factors
over Z are the same as the degrees over Q.
(Reduction modulo prime) Let f be a polynomial with integer coefficients and p a prime that does not divide the leading coefficient of f. If the reduction of f modulo p is irreducible then so
is f.
Define the field of rational functions with coefficients in F.
State a propositon on subfields of F(X).
