Abstract Algebra Problems

Transcribed Text

Denition 0.1. Let R be a ring. A non-zero element a 2 R is called a zero-divisor if there exists a non-zero element b 2 R such that a  b = 0. A ring with no zero-divisors is called an integral domain. An element a 2 R is called a unit if it has a multiplicative inverse, i.e., if there exists b 2 R such that a  b = 1. (1) Find all units and zero-divisors in the following ring: Z=18Z (2) Describe all units and zero-divisors in the ring (Z=4Z)[x], i.e., in the normal polyno- mial ring with coecients in the ring Z=4Z. (3) Show that every eld is in integral domain. (4) Show that 2n 􀀀 1 is prime only if n is prime. Is the converse true? (5) Let F be a eld. Exhibit innitely many irreducible elements in F[x], none of which is a constant multiple of another. (6) Find minimal polynomials over Q for the following algebraic numbers: * 3 + p 5 * p 2 + p 3 (7) Consider the following polynomial f(x) = x4􀀀2x3􀀀7x+14. Note that f can be con- sidered as an element of F[x], where F is one of the following elds Q;R;C. Factor f into irreducibles as an elements of F[x] in each of the three cases, i.e., when F = Q; F = R and F = C. (8) Do the same for the following polynomial x4 + 2x3 + 2x 􀀀 1, but take F to be Q; Q( p 2); R; C. (9) Consider := 2 + 4 p 7 2 Q( p 7). Find the inverse of and write it so that it is clear that it belongs to Q( p 7). (10) Consider := 1 + 1 2 3 p 2 2 Q( 3 p 2). Find the inverse of and write it so that it is clear that it belongs to Q( 3 p 2). (11) Find a polynomial f(x) of degree  2 such that x4 + x2 + 2x 􀀀 1  f(x) (mod x3 􀀀 1). (12) Recall that since x2 􀀀 2 is an irreducible element of Q[x], the ring of congruence classes Q[x]=(x2 􀀀 2)Q[x] is a eld. Find the multiplicative inverse of the following ele- ments: x + (x2 􀀀 2)Q[x]; 2 + (x2 􀀀 2)Q[x]; 3x 􀀀 1 + Q[x]: (13) Consider the following two polynomials f(x) = x2 􀀀 5 and g(x) = x2 􀀀 4x 􀀀 1. The roots of f are obviously p 5 and 􀀀 p 5. Let ; be the two roots of g. Consider the following 4 elds (why are they elds?): Q( p 5;Q(􀀀 p 5);Q( );Q( ). Are they all the same subeld of C? Prove your answer. 1

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