 # 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

## Solution Preview

This material may consist of step-by-step explanations on how to solve a problem or examples of proper writing, including the use of citations, references, bibliographies, and formatting. This material is made available for the sole purpose of studying and learning - misuse is strictly forbidden. \$30.00 for this solution

PayPal, G Pay, ApplePay, Amazon Pay, and all major credit cards accepted.

### Find A Tutor

View available Abstract Algebra Tutors

Get College Homework Help.

Are you sure you don't want to upload any files?

Fast tutor response requires as much info as possible.

SUBMIT YOUR HOMEWORK
We couldn't find that subject.
Please select the best match from the list below.

We'll send you an email right away. If it's not in your inbox, check your spam folder.

• 1
• 2
• 3
Live Chats