Modern Algebra (MATH 3163) Final Exam
NAME
(1) Suppose p is an integer other than 0, 1 such that whenever p | ab for integers
a, b, then p | a or p | b. Prove that p is prime.
(2) Let a, b, c be integers such that a = bc + 1. Prove that (a, b) = 1.
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2
(3
Modern Algebra (MATH 3163) Final Exam Solutions
(1) Suppose p is an integer other than 0, 1 such that whenever p | ab for integers
a, b, then p | a or p | b. Prove that p is prime.
Proof. Let d | p. Then p = dr for some r Z. So, p | dr. By the assumption,
HOMEWORK 2 SOLUTIONS
1.3/1a: Express 5040 as a product of primes.
Solution. It is straightforward to verify that 5040 = 24 32 5 7.
1.3/6: If p > 5 is prime and p is divided by 10, show that the remainder is 1, 3, 7, or 9.
Proof. By the Division Algorithm,
HOMEWORK 1 SOLUTIONS
1.1/1a: Find the quotient q and remainder r when a = 17 is divided by b = 4.
Solution. Put q := a/b = 4, and r := a bq = 1.
1.1/2b: a = 302, b = 19.
Solution. Put q := a/b = 15 and r := a bq = 17.
1.1/3c: Find the quotient q and remai
HOMEWORK 6 SOLUTIONS
3.3/2: Use tables to show that Z2 Z2 is isomorphic to the ring R = cfw_0, e, b, c of
Exercise 2 in Section 3.1.
Solution. The addition tables for Z2 Z2 and R are shown here:
+ 0 e b c
(0, 0) (1, 0) (0, 1) (1, 1)
+
(0, 0) (0, 0) (1, 0)
HOMEWORK 2 SOLUTIONS
1.3/1a: Express 5040 as a product of primes.
Solution. It is straightforward to verify that 5040 = 24 32 5 7.
1.3/6: If p > 5 is prime and p is divided by 10, show that the remainder is 1, 3, 7,
or 9.
Proof. By the Division Algorithm,
QUIZ 3 SOLUTIONS
(1) (T/F): One of the axioms for a ring R is that multiplication is commutative;
that is, ab = ba for all a, b R.
FALSE: This is an axiom for a commutative ring, not a general ring.
(2) (T/F): Every eld is an integral domain.
TRUE: This i
QUIZ 5 SOLUTIONS
Throughout, assume F is a eld.
(1) (T/F): There are 9 congruence classes modulo x3 + 1 in Z3 [x].
FALSE: There are 27.
(2) (T/F): If p(x) is a nonzero constant polynomial in F [x], then any two polynomials in F [x] are congruent modulo p(
QUIZ 2 SOLUTIONS
(1) (T/F): If a 3 (mod 5) and b 1 (mod 5) then 3a 2b 4 (mod 5).
FALSE: By Theorem 2.2, 3a 2b 2 (mod 5).
(2) (T/F): In Z4 , [1] = [3].
TRUE: Since 1 3 (mod 4), Theorem 2.3 implies [1] = [3].
(3) (T/F): In Zn , [a] = [b] if and only if n |
QUIZ 4 SOLUTIONS
Throughout, assume F is a eld.
(1) (T/F): If f (x) F [x], a F , and f (a) = 0, then x a divides f (x).
TRUE: This is one half of the Factor Theorem.
(2) (T/F): If f (x) F [x], a F , and x a divides f (x), then f (a) = 0.
TRUE: This is the
QUIZ 1 SOLUTIONS
(1) (T/F): If d = (a, b), there are integers u, v such that d = au + bv.
TRUE: This is the GCD theorem.
(2) (T/F): If (a, 0) = 1, then a = 1.
FALSE: a could be 1 as well.
(3) (T/F): If a | bc and (a, b) = 1, then a | c.
TRUE: This is Eucl
HOMEWORK 5 SOLUTIONS
3.2/5a: Show that a ring has only one zero element.
Proof. Suppose 0 and 0 are both zero elements in a ring R. Then
0=0+0 =0.
The rst equality followed because 0 is a zero element, the second because
0 is a zero element. Hence 0 = 0 .
NIath 3163 Section 003 Homework 1 Spring 2016
1. Prove that I3 7% cfw_21. (hint: use proof by contradiction.)
3i =ll, mm gg;r b fof
93473 lgevaJ cfw_Emu/6r
Ml 9% PM 456/5 5/ 1* (- J some.
& otuf. of WW7 (56%, )20 otjaia :5 5. f . WW"? 4) 6%
2. Let A and B
MATH 3163 Modern Algebra Midterm
NAME:
Instructions (1) Write your full name above, (2) be concise, (3) explore on scratch paper and submit
only your nal, polished, concise responses on the test, and (4) for simplicity, drop the brackets when
working in Z
HOMEWORK 11 SOLUTIONS
5.2/6: Determine the rules for addition and multiplication in Q[x]/(x2 2).
Solution. The elements of Q[x]/(x2 2) are congruence classes of polynomials of degree 1 or less in Q[x]. Let [ax + b], [cx + d] Q[x]/(x2 2).
Then
[ax + b] + [
HOMEWORK 10 SOLUTIONS
4.5/1b: Use the Rational Root Test to write f (x) = x5 + 4x4 + x3 x2 as a product
of irreducible polynomials in Q[x].
Solution. If a = r is a root of f (x), then r | (1) and s | 1. Then a = 1.
s
But neither 1 nor 1 are roots of f (x)
HOMEWORK 8 SOLUTIONS
4.2/2: If f (x) = cn xn + + c0 with cn = 0F , what is the gcd of f (x) and 0F ?
Solution. Since cn = 0F , its inverse c1 exists. Then c1 f (x) is a monic
n
n
polynomial of degree n that divides f (x) (because f (x) = cn (c1 f (x) and
HOMEWORK 9 SOLUTIONS
4.3/1b: Find a monic associate of 3x5 4x2 + 1 in Z5 [x].
Solution. Mutliply through by 31 = 2 to obtain x5 + 2x2 + 2.
4.3/3b: List all associates of 3x + 2 in Z7 [x].
Solution. The units in Z7 [x] are 1, 2, 3, 4, 5, 6. Multiplying 3x
HOMEWORK 4 SOLUTIONS
2.3/1b: Find all units in Z8 .
Solution. Recall Theorem 2.10: [a] is a unit in Zn if and only if (a, n) = 1
in Z. Since 1, 3, 5, 7 are relatively prime with 8 whereas 2, 4, 6 are not, it
follows that [1], [3], [5], [7] are the units i
HOMEWORK 13 SOLUTIONS
6.2/2: Show that every homomorphic image of a eld F is isomorphic to F itself
or to the zero ring.
Proof. First we prove that the only ideals in a eld F are (0F ) and F itself.
Indeed, let I be an ideal in a eld F . If I = (0F ), the
HOMEWORK 12 SOLUTIONS
5.3/1b: Determine whether Z5 [x]/(2x3 4x2 + 2x + 1) is a eld. Justify your answer.
Solution. Since 2 is a root of 2x3 4x2 + 2x + 1 in Z5 [x], it follows that
2x3 4x2 + 2x + 1 is reducible in Z5 [x], so Z5 [x]/(2x3 4x2 + 2x + 1) is no
Due Jan 17 Homework Set 1 Name:
(Sections 1.1)
.n_
11] = T 1
1. Use induction to prove that 1 + r + + r for n 2 1 where r at 1 is some variable.
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gumu. r to SW moJaLL [hndn dego not. {46* Ws gw
nrl. rte. 44L r1311in me {0 5:1 mm (a efwoé +0
due April 30 Homework Set 13 Name:
(section 5.3 6.2)
When writing a proof, be sure to cite all of the properties, theorems, corollaries, and denitions you use.
Be sure to write all of your answers in complete sentences (even the non-proof questions).
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