Download Review of definitions for midterm

SUMMARY OF DEFINITIONS FOR MIDTERM
BRIAN OSSERMAN
This includes all the definitions you will be expected to know, as well as the fundamental results
relating them. However, it does not include a comprehensive summary of the results we have
covered – you should be sure to look through your notes and/or Artin for these as well as for
examples.
Rather than follow the order covered in class, I have grouped the definitions by subject, while
still keeping them in a logically consistent order.
Basic definitions relating to rings
Definition. A ring R is a set with operations addition and multiplication, which are commutative
and associative, and which together are distributive. Both operations must have identity elements,
and every element must have an additive inverse.
Definition. The characteristic of a ring R is the smallest positive integer n such that
· · + 1} = 0
|1 + ·{z
n times
in R, or if no such n exists, the characteristic of R is defined to be 0.
Definition. A nonzero ring R is an integral domain if for all nonzero b, c ∈ R, the product bc is
also nonzero.
Definition. If R is a ring, a subring of R is a subset which is closed under addition, multiplication,
and subtraction, and which contains 1.
Definition. If R is a ring, and R0 ⊆ R a subring, and g1 , . . . , gn ∈ R, the ring generated by the
gi over R0 , denoted R0 [g1 , . . . , gn ] is the smallest subring of R which contains R0 together with the
gi .
Definition. An element a ∈ R is a unit if it has a multiplicative inverse.
Definition. We will say a divides b in R, and write a|b, if b is a multiple of a in R.
Definition. An integral domain R is a Euclidean domain if there exists a ‘size’ function
σ : R r {0} → Z>0
such that we can carry out division with remainder with respect to σ. That is, for any a, b ∈ R
with b 6= 0, there exist q, r ∈ R such that
a = bq + r
and either r = 0 or σ(r) < σ(b).
Definition. Let z be an element of an integral domain R. We say z is irreducible if it is not a
unit, and for any factorization z = xy in R, either x or y must be a unit. We say z is prime if z
is not a unit, and if for all x, y ∈ R such that z divides xy, then z divides x or y.
Definition. We say elements x, y ∈ R are associates if x = yu for some unit u ∈ R× .
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Definition. We say that (an integral domain) R is a unique factorization domain (or UFD) if
every nonzero, non-unit z ∈ R can be written as a product p1 · · · pn of irreducible elements, and if
further this factorization is unique, in the sense that if p01 · · · p0m is another factorization of z, then
m = n, and the p0i can be reordered so that each p0i is an associate of pi .
Definitions relating to polynomials
Definition. Let R be a ring. Given a formal symbol x, a polynomial in x with coefficients in R
is a finite formal sum
f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ,
where each ai ∈ R. The ai are called the coefficients of f (x). Polynomials of the form xi are
called monomials.
The polynomial ring R[x] is the set of all polynomials in x with coefficients in R, with the
usual rules for polynomial addition and multiplication: if f (x) is as above, and
g(x) = bm xm + bm−1 xm−1 + · · · + b1 x + b0 ,
then
max{m,n}
f (x) + g(x) =
X
(ak + bk )xk ,
k=0
where we set ak = 0 if k > n, and bk = 0 if k > m, and


m+n
X
X

f (x) · g(x) =
ai bj  xk ,
k=0
i+j=k
where we restrict i, j to be nonnegative and at most n, m respectively.
Definition. If f (x) = an xn + · · · + a0 ∈ R[x] is not equal to 0, let i be maximal so that ai 6= 0.
Then the degree of f (x) is equal to i, and the leading coefficient of f is equal to ai . We say
that f (x) is monic if the leading coefficient is equal to 1.
We say f (x) is constant if it is equal to 0, or has degree 0.
Definition. Let R be a UFD, and f (x) ∈ R[x]. We say f (x) is primitive if there is no irreducible
element p ∈ R which divides f (x).
Definitions relating to homomorphisms
Definition. A map ϕ : R → R0 is a homomorphism if ϕ(1) = 1, and for all a, b ∈ R, we have
ϕ(a + b) = ϕ(a) + ϕ(b) and ϕ(ab) = ϕ(a)ϕ(b).
Definition. A homomorphism is an isomorphism if it is bijective.
Definition. Given a homomorphism ϕ : R → R0 , the kernel is {α ∈ R : ϕ(α) = 0}.
Definition. Given ϕ : R → R0 a homomorphism, and α ∈ R0 , the homomorphism R[x] → R0
mapping x to some α and mapping r ∈ R to ϕ(r) is called the evaluation homomorphism
determined by α. If f (x) is a polynomial, we write f (α) for the image of f (x) under the homomorphism corresponding to α.
We say that α ∈ R0 is a root of f (x) if f (α) = 0.
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Definitions relating to ideals
Definition. An ideal of a ring R is a nonempty subset I such that:
(1) For all a, b ∈ I, we have a + b ∈ I.
(2) For all a ∈ R and b ∈ I, we have ab ∈ I.
Definition. If a ∈ R, we have the ideal consisting of the multiples of a. This is denoted (a), or
aR. Ideals of this form are called principal ideals.
The ideal which is all of R is called the unit ideal.
The principal ideal (0) contains only 0, and is called the zero ideal.
An ideal is called proper if it is not the unit ideal or the zero ideal.
Definition. A principal ideal domain (or PID) is an integral domain such that every ideal is
principal.
Definition. Given a1 , . . . , am ∈ R, the ideal generated by a1 , . . . , am is the smallest ideal of R
containing all the ai .
Definition. Let F be a field, and f, g ∈ F [x] polynomials, not both equal to 0. The greatest
common divisor d ∈ F [x] of f and g is the unique monic polynomial which generates the ideal
(f, g).
Definition. An ideal I ( R is maximal if the only ideal of R strictly containing I is the unit
ideal.
Definition. An ideal I ( R is prime if, for all a, b ∈ R such that ab ∈ I, either a ∈ I or b ∈ I.
Definition. Let I, J be ideals of a ring R. Then the product ideal IJ is the ideal generated by
elements of the form ab, with a ∈ I and b ∈ J. Explicitly,
n
X
IJ = {
ai bi : ai ∈ I, bi ∈ J}.
i=1
Definition. Given a ring R and an ideal I, the quotient ring R/I is the ring structure on the
set of cosets of I in R induced by addition and multiplication in R.
Constructions with rings
Definition. Let f (x) ∈ R[x] be a nonzero polynomial. To adjoin an element α to R which is a
root of f (x), set
R[α] := R[x]/(f (x)),
where α denotes the congruence class of x.
Definition. Given β ∈ R, to adjoin an inverse of β to R we take the ring R[α] := R[x]/(βx − 1).
Definition. Let R be an integral domain. A fraction is a symbol a/b or ab with a, b ∈ R, and
b 6= 0. Two fractions a/b and c/d are equivalent if ad = bc. We can add and multiply fractions as
follows:
a c
ad + bc
a c
ac
+ =
,
· = .
b d
bd
b d
bd
The set of equivalence classes of fractions in R is the fraction field of R.
Definition. The fraction field of a polynomial ring K[x] over a field K is called the field of
rational functions with coefficients in K, and written K(x).
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Definitions relating to field extensions
Definition. Let F be a subfield of a field K, and α ∈ K. We say α is algebraic over F if there
exists a nonzero polynomial f (x) ∈ F [x] such that f (α) = 0. If α is algebraic, then the evaluation
homomorphism F [x] → K sending g(x) ∈ F [x] to g(α) has a nonzero kernel. But since F [x] is a
PID, this kernel is principal, say (h(x)) for some monic polynomial h(x) ∈ F [x]. We call h(x) the
minimal polynomial of α over F .
Definition. α ∈ C is an algebraic integer if it is algebraic over Q, and the coefficients of its
minimal polynomial are in Z.
Definition. Let α ∈ C be algebraic. The subset R of algebraic integers in Q(α) is called the ring
of integers of Q(α).
Definition. A field extension of a field F is field K together with an inclusion of F as a subfield
of K. Such an extension will be denoted K/F .
Definition. We say a field extension K/F is algebraic if every α ∈ K is algebraic over F .
We say α ∈ K is transcendental over F if α is not algebraic over F .
If α ∈ K is algebraic over F , we define the degree of α over F to be the degree of the minimal
polynomial of α over F .
Definition. Given a field extension K/F , and an element α ∈ K, the field generated by α over
F , denoted F (α), is the smallest subfield of K containing F and α. Similarly, given α1 , . . . , αn ∈ K,
then F (α1 , . . . , αn ) is the smallest subfield of K containing F and all the αi .
Definition. If K/F is a field extension such that K is a finite-dimensional vector space over F ,
then we say that K/F is a finite extension, and define the degree of K/F to be dimF K. We
write [K : F ] for the degree of K over F .
Basic results relating the definitions
Proposition. Let R be an integral domain. Then every prime element of R is irreducible.
If every irreducible element of R is prime, then factorizations into irreducibles are unique whenever they exist. Conversely, if R is a UFD, then every irreducible element is prime.
Proposition. Given an integral domain R and x, y ∈ R, we have that x is an associate of y if and
only if x|y and y|x, if and only if (x) = (y).
Proposition. For any ideal I of R, we have R/I a field (respectively, an integral domain) if and
only if I is a maximal ideal (respectively, a prime ideal).
Corollary. A maximal ideal is prime.
Theorem. A principal ideal domain is a unique factorization domain.
Proposition. Every Euclidean domain is a principal ideal domain, and hence a unique factorization
domain.
Theorem. If R is an integral domain, then R[x] is an integral domain. If R is a UFD, then R[x]
is a UFD.
Proposition. Given a field extension K/F , if α is algebraic over F , then F (α) = F [α] ∼
=
F [x]/(f (x)), where f (x) is the minimal polynomial of α over F .
In this case, the degree of α over F is equal to [F (α) : F ].
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