MATH 104A Homework Assignments (Winter, 2010)
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HW7   Due by 5 pm Friday March 12

page 108 #1-6

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HW 6  Due by 5 pm Friday March 5 

Let p be congruent to 1 or 3 mod 8, p prime.
Show that p has the form a^2 + 2b^2.

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HW 5 Due by 5 pm Friday February 19

p. 84 #1-8

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HW 4 Due by 5 pm Friday February 12

page 62:  #1, 5, 6, 7
page 69:  #1, 2, 3, 4, 5

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HW 3   Due by 5pm Friday January 22

For brevity, let s = sqrt(-5).
(A)  Show that there is no unique factorization theorem for the ring Z[s].
Hint:  Use the equality  9 = 3*3 = (2+s)*(2-s) and show that
3 and (2+s) are non-associated primes in Z[s].
Note that the only units in Z[s] are 1 and -1.
(B)  Show that in Z[s],  1 is a gcd of 3 and (2+s).
(C)  Jane says that because of problem (B), 
some linear combination of 3 and (2+s)  (with coefficients in Z[s])
is equal to 1.  Prove that Jane is wrong.
(D)  Show that gcd( 9, 6+3s) does not exist in Z[s].
Hint:  Show that if a gcd were to exist, it would have to be 2+s (or -2-s).  
Then get a contradiction
by showing that 3 does not divide 2+s in the ring Z[s].
Supplementary hint:  2+s is clearly a common divisor, 
so a greatest common divisor, if it existed, would be
some multiple of 2+s, such as (2+s)*(a+bs).  But one can show that (2+s)*(a+bs) cannot 
be a common divisor unless a+bs is 1 or -1, ergo (2+s) is a gcd, if a gcd were to exist.
(E)  Show that there is no unique factorization theorem for the ring Z[t],
where t = sqrt(-13).  Hint:  Cf. problem (A).
(F)  Let a,b,c,e  be integers with gcd(a,b,c) = 1.  Prove that the Diophantine equation
ax+by+cz = e has infinitely many solution triples x,y,z  (with integers x, y, z).
(G)  Find all (integer) solution pairs x, y  to the Diophantine equation  5x + 7y = 11.

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HW2  Due by 5pm Friday  January 15

Problem A:  Let s denote the square root of 10.
Prove that 2,3, 4+s, and 4-s are each prime in the ring
{a+bs : a, b are integers}.

page 34:  #2-4, 10-11
page 39:  #7, 10, 13

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HW1  Due by 5pm Friday  January  8

page 7, #1,2
page 17, #1,2
page 24, #9