代數學 Theory of Rings

傲月光希

1.Let　R be a commutative ring with identity.
　(a)Let　I and　J be ideals of　R. Prove that　I+J is the ideal generated by　I∪J
　(b)If　I+J=R, prove that　I∩J=IJ.
　(c)Let　R=Z24,I=<[4]> and　J=<[6]>. Find　I+J,I∩J, and　IJ.

2.Let　T={a/b |a/b belongs to　Q, a and b are relatively prime and 5 does not divide b}
(a)Prove that　T is a ring under usual addition and multiplication.
(b)Identify the unit group of　T.
(c)Find a complete set of irreducible elements of　T.
(d)Prove that　I = {a/b |5 divides a} is an ideal of　T and the quotient ring　T/I is a field.
(e)Show that　T is a PID.

3.Let　F be a field and　F[x,y] be the polynomial ring in variables　x and　y.
(a)Prove that the ideal <x> is a prime ideal but not maximal.
(b)Find a maximal ideal that contains the ideal <x>.
(c)Prove that　F[x,y] is　not a PID.

4.Answer the following true-or-false questions. If true, prove it; otherwise, disprove it.
(a)Every field is a PID.
(b)Let　R be a commutative ring with identity. Then {0} is a maximal ideal if and only if　R is a field.
(c)Let　θ:Zp →R be a non-zero ring homomorphism where p is a prime. Then　Zp is isomorphic some subring of　R.
(d)A factor ring of an integral domain is still an integral domain.
(e)Any subgroup of the multiplicative group　C* is cyclic.

5.Prove that in a commutative ring with identity, every nonzero prime ideal of finite index is maximal.

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Symbols Explanation:
Z=the set of all integers.
Q=the set of all rational numbers.
Zn=the set of remainders which is of a integer divided by n.(e.g.Z2={0,1})

1.
(a) List All irreducible polynomials of degree less than 4 over　Z2.
(b) Show that　f(x)=(x^4)+x+1 is irreducible over　Z2.
(c) Contruct a finite field　K of order 16 by using　f(x) above. Let α=x+<f(x)> in　K.
　　Express the general form of elements of　K in terms of α.
(d) Realize the additive group <K,+>.(It means what group is isomorphic to <K,+>)
(e) Is　K*=K-{0} cyclic under multiplication? If so, find a generator of　K*.
(f) Find all the roots of　f and show that　K is a splitting field of　f over　Z2.

2. Test whether the following polynomials are irreducible over　Q.
(a)　f1(x)=(x^5)+9(x^4)+12(x^2)+6.
(b)　f2(x)=8(x^3)-6x+1.
(c)　f3(x)=(x^6)+(x^3)+1. (HINT: Use a suitable change of variable.)
(d)　f4(x)=(3/7)(x^4)-(2/7)(x^2)+(9/35)(x)+3/5.

3.Show that 3(x^2)+4x+3 in　Z5[x] factors as (3x+2)(x+4) and (4x+1)(2x+3).
　Can you conclude that　Z5[x] is　not a UFD? Explain your answer.

4.
(a) Find the splitting field　F of　f(x)=(x^4)-2 over　Q.
(b) Find the degree (F :Q).
(c) Find a basis ofF over　Q.
(d) Write down two distinct subfields of degree 2 over　Q between　F and　Q.

5. Answer the following true-or-false questions. If True, prove it;otherwise, disprove it.
(a)　Z[x,y] is a UFD.
(b)   as rings.
(c) A simple algebraic extension must be a finite extention.
(d) If　F is an infinite field, then the characteristic of　F must be 0.

[ 本文最後由 傲月光希 於 07-6-16 10:18 PM 編輯 ]