Download MAT 236, elementary Abstract Algebra Some Important Definitions

MAT 236, elementary Abstract Algebra
Some Important Definitions and Theorems
Z is the set of integers.
Q is the set of rational numbers
R is the set of real numbers
C is the set of complex numbers.
Well-Ordering Axiom. If S is a subset of nonnegative integers and S is nonempty, then S has
a smallest element.
The Division Algorithm. Let a, b be integers with b > 0. Then there exist unique integers q
and r such that:
a = bq + r
AND
0 ≤ r < b.
Definition: Let a, b ∈ Z with b 6= 0. We say that b divides a (or that b is a divisor of a, or
that b is a factor of a) if a = bq for some integer q.
Notation: If b divides a, we write b|a. So:
b|a ⇔ a = bq for some q ∈ Z.
Definition: Let a, b ∈ Z with not both 0. The greatest common divisor (gcd) of a and b is
a (positive) integer d such that:
1. d|a and d|b;
2. if c ∈ Z such that c|a and c|b, then c ≤ d.
We use (a, b) or gcd(a, b) to denote the gcd of a and b.
Theorem 1.3: Let a, b ∈ Z, not both zero. Let d = (a, b). Then there exist integers u, v such
that d = au + bv. (Note, in general the integers u and v are not unique.) Furthermore, d is the
smallest positive integer that can be written in the form ax + by for some x, y ∈ Z.
NOTE:(1) Just because an integer can be written in the form ax + by does NOT mean the
integer is the gcd.
(2) There are typically an infinite number of integers u and v such that (a, b) = au + bv.
Definition: Let a, b ∈ Z with not both 0. We say a and b are relatively prime if (a, b) = 1.
Corollary 1.3a: Let a, b ∈ Z, not both zero, and assume there exist integers x and y such that
ax + by = 1. Then (a, b) = 1.
Theorem 1.5: If a, b, c ∈ Z such that a| bc AND (a, b) = 1, then a| c.
Definition: An integer p is prime if p =
6 0, ±1 and the only divisors of p are ±1 and ±p. An
integer q is composite if q 6= 0, ±1 and q is not prime.
Euclid’s Lemma (Theorem 1.8): Let p be an integer with p 6= 0, ±1. Then p is prime if and
only if it has the property:
for any integers b and c, if p| bc then p| b or p| c.
The Fundamental Theorem of Arithmetic (Theorem 1.11): Every integer n except 0, ±1
is a product of primes. This prime factorization is unique in teh following sense: If
n = p1 p2 · · · pr and n = q1 q2 · · · qs
with each p1 , qj a prime number, then r = s and after reordering and relabeling the q’s,
p1 = ±q2 , p2 = ±q2 , . . . , pr = ±qr .
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Definition: Let S be a set. A relation on the set S is a subset R of all ordered pairs of elements
in S. (We denote the set of all ordered pairs of elements in S by S × S.) Many times, instead of
writing (a, b) ∈ R, we write a ∼ b to denote the relation ∼ that is defined by R.
Definition: Let ∼ be a relation on a set S. Then ∼ is an equivalence relation if it has the
following properties:
1. (Reflexive) For all a ∈ S we have a ∼ a.
2. (Symmeteric) For all a, b ∈ S we have a ∼ b ⇒ b ∼ a.
3. (Transitive) For all a, b, c ∈ S we have a ∼ b and b ∼ c ⇒ a ∼ c.
Definition: Let ∼ be an equivalence relation on a nonempty set S and let a ∈ S. The equivalence class of a is the set
[a] = {b ∈ S : a ∼ b}.
Theorem D.1: Let ∼ be an equivalence relation on a nonempty set S and let a, b ∈ S. Then
[a] = [b] if and only if a ∼ b.
Definition: Let S be a nonempty set. A partition of S is a collection of subsets P such that:
1. for any a ∈ S there exists a U ∈ P such that a ∈ U .
2. for any two distinct subsets U, V ∈ P, we have U ∩ V = ∅.
Corollary D.2a: Let ∼ be an equivalence relation on a nonempty set S. Then the equivalence
classes of ∼ form a partition of S. Conversely, if P is a partition of S, then we can define an
equivalence relation∼ on S where the sets in P are the equivalence classes of ∼.
Definition: Let n ∈ Z be such that n ≥ 2. Then for any two integers a, b ∈ Z we say a is
congruent to b modulo n if n| (a − b). In this case we write a ≡ b (mod n). So:
a ≡ b (mod n) ⇔ n| (a − b).
Example/Theorem 2.1: Let n ≥ 2 be an integer. Then congruence modulo n is an equivalence
relation on Z.
Definition: Let n ∈ Z such that n ≥ 2. For any a ∈ Z, the congruence class of a modulo n
is the set
[a] = [a]n = {b ∈ Z : a ≡ b (mod n)}.
(Note this is just the equivalence class of a under the equivalence relation ≡ (mod n).) The
collection of all congruence classes modulo n is denoted by Zn or Z/(nZ).
Theorem 2.3/Corollary 2.4: Let n ∈ Z with n ≥ 2. Then Zn is a partition of Z.
(Note: This follows directly from Theorem 2.1 and Corollary D.2.)
Corollary 2.5: Let n ≥ 2 be an integer and consider congruence modulo n.
1. For any a ∈ Z, we have a ≡ r (mod n) where r is the remainder of a when divided by n.
2. There are exactly n congruence classes in Zn . In particular,
Zn = {[0], [1], [2], . . . , [n − 1]}.
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Theorem 2.2: Let n ∈ Z with n ≥ 2. For any a, b, c, d ∈ Z, if a ≡ b (mod n) and
c ≡ d (mod n), then
1. a + c ≡ b + d (mod n);
2. ac ≡ bd (mod n).
Theorem 2.8: Let p ≥ 2 be an integer. Then the following conditions are equivalent:
1. p is prime
2. For any a 6= 0 in Zp , the equation ax = 1 has a solution in Zp . (Such a solution u is called
the multiplicative inverse of a and is denoted by a−1 ∈ Zp .)
3. Whenever ab = 0 in Zp , the a = 0 or b = 0 in Zp .
Fermat’s Little Theorem: Let p ≥ 2 be a prime and let a ∈ Z. If p does not divide a then
ap−1 ≡ 1 (mod p). More generally, for any a ∈ Z we have ap ≡ a (mod p).
Theorem 12.3: Let p, q be distinct positive primes. Let n = pq and k = (p − 1)(q − 1). Choose
1 < e < k such that (e, k) = 1. Then there exists a 1 < d < k such that ed ≡ 1 (mod k) and for
any M ∈ Z we have
(M e )d ≡ M (mod n).
Definition: Let S and T be two nonempty sets. The Cartesian Product of S and T , denoted
S × T is the set of ordered pairs (s, t) such that s ∈ S and t ∈ T . So
S × T = {(s, t) : s ∈ S, t ∈ T }.
Definition: A function f from a set S to a set T is a rule or procedure that unambiguously
assigns to every element of S an element of t. We write f : S → T to denote such a function f .
Furthermore, for any s ∈ S, we let f (s) denote the element of T that gets assigned to s.
In this case, we say S is the domain of f , T is the codomain of f , and the subset of T of
all elements that get assigned to an element of S (i.e., the set {t ∈ T : t = f (s) for some s ∈ S})
is calle the image of f or the range of f .
Definition: A binary operation on a nonempty set S is a function b : S × S → S. For any
s1 , s2 ∈ S, we typically denote the element b(s1 , s2 ) by s1 ∗ s2 for some corresponding symbol ∗.
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Definition: A ring is a nonempty set R equipped with two binary operations (usually written
as + and ×) that satisfy the following axioms. For all a, b, c ∈ R:
1. (closure of addition) If a, b ∈ R then a + b ∈ R.
2. (associative addition) a + (b + c) = (a + b) + c.
3. (commutative addition) a + b = b + a.
4. (additive identity) There exists an element 0R ∈ R such that for any a ∈ R, a + 0R = a =
0R + a.
5. (additive inverse) For each a ∈ R, there exists an element a0 ∈ R such that a + a0 = 0R .
6. (closure of multiplication) If a, b ∈ R then a × b ∈ R.
7. (associative multiplication) a × (b × c) = (a × b) × c.
8. (distributive laws) a × (b + c) = a × b + a × c AND (a + b) × c = a × c + b × c.
When the binary operation × is understood, we will abbreviate a × b by ab.
A ring R is said to be a commutative ring if, in addition to the above eight axioms, it also
satisfies the axiom:
9. (commutative multiplication) ab = ba for all a, b ∈ R.
A ring R is a ring with identity if, in addition to the original eight axioms, it also contains an
element 1R satisfying
10. (multiplicative identity) For all a ∈ R we have 1R a = a = a1R .
A commutative ring with identity (satisfying Axioms 1 - 10) is an integral domain if 1R 6= 0R
AND
11. Whenever a, b ∈ R and ab = 0, then a = 0 or b = 0.
A commutative ring with identity (satisfying Axioms 1 - 10) is a field if 1R 6= 0R AND
12. For each a 6= 0R in R, there is an a0 ∈ R such that aa0 = 1R = a0 a.
Theorem 3.1: Let R and S be rings. Define addition and multiplication on the Cartesian
Product R × S for any (r, s), (r0 , s0 ) ∈ R × S by
(r, s)+(r0 , s0 ) = (r + r0 , s+s0 ) and (r, s) ? (r0 , s0 ) = (r · r0 , s ∗ s0 ).
With this addition (+) and multiplication (?), R × S is a ring. If R and S are both commutative,
so is R × S. If both R and S have an idenity, then so does R × S.
Definition: Let R be a ring. A subring of R is a subset S of R that is also a ring using the
addition and multiplication of R.
Theorem 3.2: Suppose that R is a ring and that S is a subset of R. Then S is a subring of R
if and only if
1. 0R ∈ S.
2. For all a, b ∈ S we have a + b ∈ S.
3. For all a, b ∈ S we have ab ∈ S.
4. For evey a ∈ S, the solution to the equation a + x = 0R is in S.
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Lemma 3.3: Let R be a ring.
1. The zero element of R is unique and denoted 0R .
2. For any a ∈ R, the solution to a + x = 0R in R is unique and is denoted −a.
3. If R has an identity, it is unique and denoted 1R .
4. If R has an identity and a ∈ R is such that the equations ax = 1R and ya = 1R , have
solutions, then these solutions are unique and equal. We denote this solution by a−1 .
Theorem 3.4: Let R be a ring and let a, b, c ∈ R. If a + b = a + c, then b = c.
Theorem 3.6: Let R be a ring and let S ⊆ R be a subSET. Then S is a subRING of R if and
only if
(0) S is nonempty.
(1) S is closed under subtraction.
(2) S is closed under multiplication.
Notation: Let R be a ring, a ∈ R and let n ∈ Z with n ≥ 1. Then
• na = a + a + · · · + a(n times)
• 0a = 0R
• −na = (−a) + (−a) + · · · + (−a)(n times).
NOTE: When we write 5a we are NOT implying that 5 is in the ring R. Instead we just mean
we are adding a to itself 5 times. Still, if 5 is an element of R, these end up being the same thing.
Definition: Let R be a ring with identity. A unit in R is an element u ∈ R such that the
equations ux = 1R and yu = 1R both have solutions in R. If u is a unit in R, there is a unique
solutino u−1 to both these equation by Lemma 3.3. We call u−1 the multiplicative inverse of
u.
Theorem 3.9: Every field is an integral domain.
Theorem 3.10: Let R be an integral domain. If 0R 6= a ∈ R and ab = ac for some b, c ∈ R,
then b = c.
Theorem 3.11: Every finite integral domain is a field.
Definition: Let R be a ring and let a ∈ R. Then a is a zero divisor in R if a 6= 0R AND there
exists a b 6= 0R in R such that ab = 0R or ba = 0R .
Definition (informal): A function f from a set S to a set T (denoted f : S → T ) is a rule or
procedure that unambiguously assigns exactly one element of T to each element of S. The set S
is the domain of f , the set T is called the codomain of f , and the set f (S) = {f (s) : s ∈ S}
is called the range or image of f .
Definition: Let S, T, U be sets and let f : S → T and g : T → U be functions. The composite
g ◦ f of f and g is the function from S to U defined by the formula (g ◦ f )(x) = g(f (x)) for all
x ∈ S. In terms of diagrams, we have
f
g
g ◦ f : S −→ T −→ U.
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Definition: Let f : S → T be a function.
• The function f is one-to-one or injective if for any a, b ∈ S we have f (a) = f (b) ⇒ a = b.
• The function f is onto or surjective if for any t ∈ T , we can find at least one s ∈ S such
that t = f (s).
• The function f is a one-to-one correspondence or bijective if it is both one-to-one and
onto.
Definition: Let S be a set. The identity function on S is the function ιS : S → S defined
by ιS (x) = x for all x ∈ S.
A function f : S → T is invertible if there exists a function g : T → S such that
g ◦ f = ιS
AND
f ◦ g = ιT .
Theorem B.1: A function f : S → T is invertible if and only if it is both one-to-one and onto.
Definition: Let R and S be rings and let φ : R → S be a function. Then φ is a ring
homomorphism if, for all a, b ∈ R we have:
(1) φ(a + b) = φ(a) + φ(b)
and
(2) φ(ab) = φ(a)φ(b).
We say that φ is a ring isomorphism if φ satisfies conditions (1) and (2) above AND
(3) φ is one-to-one
and
(4) φ is onto.
We say that two rings R and S are isomorphic if there exists an isomorphism φ : R → S. In
this case we write R ∼
= S.
Theorem 3.12: Let φ : R → S be a homomorphism of rings. Then
(1) φ(0R ) = 0S .
(2) φ(−a) = −φ(a) for every a ∈ R.
(3) φ(a − b) = φ(a) − φ(b) for every a, b ∈ R.
If R is a ring with identity and φ is onto, then
(4) S is a ring with identity and φ(1R ) = 1S .
(5) Whenever u is a unit in R, then φ(u) is a unit in S and φ(u)−1 = φ(u−1 ).
Theorem 3.14: Let R, S, and T be rings. Then
(1) R ∼
= R.
(2) If R ∼
= S then S ∼
= R.
∼
∼
(3) If R = S and S = T , then R ∼
= T.
Definition: Let φ : R → S be a homomorphism of rings. Then the image of φ is the set
Im(φ) = φ(R) = {s ∈ S : s = φ(r) for some r ∈ R}.
The kernel of φ is the set
ker(φ) = {r ∈ R : φ(r) = 0S }.
Corollary 3.13: If φ : R → S is a homomorphism of rings, then Im(φ) is a subring of S and
ker(φ) is a subring of R.
Theorem 6.11: Let φ : R → S be a homomorphism of rings. Then φ is one-to-one if and only
if ker(φ) = {0R }. (Note: ker(φ) = 0R means that the only solution to φ(x) = 0S is x = 0R .)
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Theorem 4.1: If R is a ring, then there exists a ring P that contains R and an element x that
is not in R and P has the following properties:
(1) R is a subring of P .
(2) xa = ax for all a ∈ R.
(3) Every element of P can be written in the form
a0 + a1 x + a2 x2 + · · · + an xn for some n ≥ 0 and ai ∈ R.
(4) If n ≤ m and
a0 + a1 x + · · · + an xn = b0 + b1 x + · · · + bm xm ,
for some ai , bi ∈ R, then ai = bi for all 0 ≤ i ≤ n and bi = 0R for all i > n.
(5) a0 + a1 x + · · · + an xn = 0R if and only if ai = 0R for all i.
(See Appendix G for such a construction.)
Definition: If R is a ring, we denote the ring P in Theorem 4.1 by R[x] and we call this ring
the ring of polynomials in the variable x with coefficients in R.
Definition: Let R be a ring and let
a0 + a1 x + · · · + an xn , b0 + b1 x + . . . + bm xm ∈ R[x]
with n ≤ m. Then
(a0 + a1 x + · · · + an xn ) + (b0 + b1 x + . . . + bm xm )
= (a0 + b0 ) + (a1 + b1 )x + · · · + (an + bn )xn + bn+1 xn+1 + · · · + bm xm
and
(a0 + a1 x + · · · + an xn )(b0 + b1 x + . . . + bm xm ) = c0 + c1 x + · · · + cn+m xn+m
where for each j,
cj = a0 bj + a1 bj−1 + a2 bj−1 + · · · + aj b0 =
j
X
ai bj−i .
i=0
(NOTE in the above notation, if i < 0 or i > m, we define bi = 0R . Similarly with ai .)
Definition: Let f (x) = a0 + a1 x + · · · + an xn ∈ R[x] for some ring R. If an 6= 0R , we say that
an is the leading coefficient of f (x). In this case, we say f (x) has degree n and we write
deg f (x) = n.
If R is a ring with identity, we say f (x) ∈ R[x] is monic if it is nonzero and its leading
coefficient is 1R .
Theorem 4.2: If R is an integral domain and f (x), g(x) are nonzero polynomials in R[x], then
deg[f (x)g(x)] = deg f (x) + deg g(x).
Corollary 4.3: If R is an integral domain, then so is R[x].
The Division Algorithm for Polynomials: Let F be a field and f (x), g(x) ∈ F [x] with
g(x) 6= 0. Then there exists unique polynomials q(x) and r(x) such that
f (x) = g(x)q(x) + r(x) AND either r(x) = 0F or deg(r(x)) < deg(g(x)).
Definition: Let F be a field and let f (x), g(x) ∈ F [x] with f (x) nonzero. The f (x) divides
g(x) (and we write f (x) | g(x)) if g(x) = f (x)q(x) for some q(x) ∈ F [x].
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Definition: Let F be a field and f (x), g(x) ∈ F [x], not both zero. The greatest common
divisor of f (x) and g(x) is the monic polynomial d(x) such that
(1) d(x) | f (x) and d(x) | g(x)
AND
(2) if c(x) ∈ F [x] such that c(x) | f (x) and c(x) | g(x), then deg c(x) ≤ deg d(x).
If (f (x), g(x)) = 1F , we say f (x) and g(x) are relatively prime.
Theorem 4.5: Let F be a field and let f (x), g(x) ∈ F [x], not both zero. Then the greatest
common divisor d(x) = (f (x), g(x)) is the monic polynomial of smallest degree that can be
written in the form
d(x) = f (x)u(x) + g(x)v(x) for some u(x), v(x) ∈ F [x].
Theorem 4.7: Let F be a field and f (x), g(x), h(x) ∈ F [x]. If f (x) | g(x)h(x) and (f (x), g(x)) =
1F , then f (x) | h(x).
Definition: Let F be a field and let f (x), g(x) ∈ F [x]. We say f (x) and g(x) are associates in
F [x] if g(x) = cf (x) for some nonzero c ∈ F .
Definition: Let F be a field and let f (x) ∈ F [x]. We say f (x) is irreducible in F [x] if f (x)
is not a constant polynomial and its only divisors in F [x] are constants and associates of f (x)
in F [x]. A reducible polynomial in F [x] is a nonconstant polynomial g(x) ∈ F [x] that is not
irreducible.
Observation: If F is a field, then any degree 1 polynomial in F [x] is irreducible. (Note, the
converse is NOT true in general.)
Theorem 4.10: Let F be a field. A nonzero polynomial f (x) ∈ F [x] is reducible in F [x] if it
can be written as the product of two polynomials in F [x] of lower degree.
Theorem 4.11: Let F be a field and let p(x) ∈ F [x] be nonconstant. Then the following are
equivalent:
(1) p(x) is irreducible in F [x].
(2) If b(x) and c(x) are any polynomials in F [x] such that p(x) | b(x)c(x), then p(x) | b(x) or
p(x) | c(x).
(3) If r(x), s(x) ∈ F [x] are such that p(x) = r(x)s(x), then r(x) or s(x) is a nonzero constant
polynomial.
Theorem 4.13: Let F be a field. Every nonconstant polynomial f (x) ∈ F [x] is a product of
irreducible polynomials in F [x] (where we allow products to consist of only one factor). This
factorization is unique in the sense that if
f (x) = p1 (x)p2 (x) · · · pr (x)
and
f (x) = q1 (x)q2 (x) · · · qs (x)
with each pi (x) and qj (x) irreducible in F [x], then r = s and after the qj (x) are reordered and
relabeled if necessary, for each i
pi (x) is an associate of qi (x).
Definition: Let R be a ring and let f (x) ∈ R[x] be given by
f (x) = cn xn + cn−1 xn−1 + · · · + c1 x + c0 , for cn , cn−1 , . . . , c1 , c0 ∈ R.
Then for any a ∈ R we define
f (a) = cn an + cn−1 an−1 + · · · + c1 a + c0 ∈ R.
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In this way, the polynomial f (x) induces a polynomial function from R to R given by a 7→ f (a).
Definition: Let R be a ring and let f (x) ∈ R[x]. An element a ∈ R is a root of f (x) if
f (a) = 0R .
The Remainder Theorem: Let F be a field, f (x) ∈ F [x], and a ∈ F . Then f (a) is equal to
the remainder when f (x) is divided by x − a.
The Factor Theorem: Let F be a field, f (x) ∈ F [x], and a ∈ F . Then a is a root of the
polynomial f (x) if and only if (x − a) is a factor of f (x).
Corollary 4.16: Let F be a field and let f (x) ∈ F [x] be nonzero of degree n. Then f (x) has at
most n roots in F .
Corollary 4.18: Let F be a field and let f (x) ∈ F [x] be a polynomial such that
deg(f (x)) = 2
or
deg(f (x)) = 3.
Then f (x) is irreducible in F [x] if and only if f (x) has no roots in F .
(NOTE: If deg(f (x)) ≥ 4, you CANNOT use this corollary.)
Rational Root Test: Let f (x) = an xn + an−1 xn−1 + · · · + a1 x + a0 ∈ Z[x]. If r, s ∈ Z are
nonzero and the rational number r/s is a root of f (x) in lowest terms,then r | a0 and s | an .
Lemma 4.21: Let f (x), g(x), h(x) ∈ Z[x] with f (x) = g(x)h(x). If p ∈ Z is a prime that
divides every coefficient of f (x), then either p divides every coefficient of g(x) or p divides every
coefficient of h(x).
Theorem 4.22: Let f (x) ∈ Z[x]. then f (x) factors as a product of polynomials of degrees m
and n in Q[x] if and only if f (x) factors as a product of polynomials of degrees m and n in Z[x].
Eisenstein’s Criterion: Let f (x) = an xn + · · · + a1 x + a0 ∈ Z[x] be nonconstant. If there is a
prime p ∈ Z such that:
(1) p divides each of a0 , a1 , . . . , an−1
(2) p does not divide an
(3) p2 does not divide a0 ,
then f (x) is irreducible in Q[x].
Theorem 4.24: Let f (x) = ak xk + · · · + a1 x + a0 ∈ Z[x], and let p ∈ Z be a prime such that p
does not divide ak . If f¯(x) = [ak ]p xk + · · · + [a1 ]x + [a0 ] ∈ Zp [x] is irreducible in Zp [x], then f (x)
is irreducible in Q[x].
Definition: Let F be a field and p(x) ∈ F [x] be nonconstant. Then for any f (x), g(x) ∈ F [x]
we say f (x) is congruent to g(x) modolu p(x) if p(x) | [f (x) − g(x)]. In this case we write
f (x) ≡ g(x) (mod p(x)).
Theorem 5.1: Let F be a field and let p(x) ∈ F [x] be nonconstant. Then congruence modulo
p(x) is an equivalence relation.
Definition: Let F be a field and let p(x) ∈ F [x] be nonconstant. For any f (x) ∈ F [x], the
congruence class of f (x) modulo p(x) is the set
[f (x)] = {g(x) ∈ F [x] : f (x) ≡ g(x) (mod p(x))} .
The set of all congruence classes modulo p(x) is denoted by
F [x]/(p(x)) or F [x]/hp(x)i.
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Corollary 5.5: Let F be a field and let p(x) ∈ F [x] be nonconstant of degree n. Then
F [x]/(p(x)) = {[g(x)] : deg(g(x)) < n or g(x) = 0} .
Furthermore, if g(x), h(x) ∈ F [x] are either zero or have degree less than n, then [g(x)] = [h(x)]
if and only if g(x) = h(x).
Theorem 5.2/5.6 Let F be a field and let p(x) ∈ F [x] be nonconstant. Let f (x), g(x), h(x), k(x) ∈
F [x] be such that f (x) ≡ g(x)(mod p(x)) and h(x) ≡ k(x)(mod p(x)) so that [f (x)] = [g(x)] and
[h(x)] = [k(x)]. Then
(a) f (x) + h(x) ≡ g(x) + k(x)(mod p(x)) or, equivalently, [f (x) + h(x)] = [g(x) + k(x)]
(b) f (x)h(x) ≡ g(x)k(x)(mod p(x)) or, equivalently, [f (x)h(x)] = [g(x)k(x)].
As a result, we can define an addtion and multiplication on F [x]/(p(x)) by
[f (x)] + [h(x)] = [f (x) + h(x)] and [f (x)][h(x)] = [f (x)h(x)]
Theorem 5.7/5.8: Let F be a field and let p(x) ∈ F [x] be nonconstant. Then F [x]/(p(x)) is a
commutative ring with identity that contains F as a subring.
Theorem 5.9’: Let F be a field and let p(x), f (x) ∈ F [x] with p(x) nonconstant and f (x) 6≡ 0
(mod p(x)).
(a) [f (x)] is a unit in F [x]/(p(x)) if and only if (f (x), p(x)) = 1.
(b) [f (x)] is a zero divisor in F [x]/(p(x)) if and only if (f (x), p(x)) 6= 1.
Theorem 5.10: Let F be a field and let p(x) ∈ F [x] be nonconstant. Then F [x]/(p(x)) is a
field if and only if p(x) is irreducible in F [x].
Theorem 5.11: Let F be a field and let p(x) ∈ F [x] be irreducible in F [x]. Then the field
E = F [x]/(p(x)) contains a root of p(y) ∈ E[y] In particular, [x] ∈ E is a root of p(y) ∈ E[y].
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