Call Now to Set Up Tutoring:

(888) 888-0446

Abstract Algebra : Splitting Fields

Study concepts, example questions & explanations for Abstract Algebra

Create An Account

All Abstract Algebra Resources

9 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Fields

What definition does the following correlate to?

If $\displaystyle b$ is a prime, then the following polynomial is irreducible over the field of rational numbers.

$\phi(x)=x^{p-1}+...+x+1$ $\displaystyle \phi(x)=x^{p-1}+...+x+1$

Possible Answers:

Eisenstein's Irreducibility Criterion

Primitive Field Theorem

Principal Ideal Domain

Ideals Theorem

Gauss's Lemma

Correct answer:

Eisenstein's Irreducibility Criterion

Explanation:

The Eisenstein's Irreducibility Criterion is the theorem for which the given statement is a corollary to.

The Eisenstein's Irreducibility Criterion is as follows.

$f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$ $\displaystyle f(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0$

is a polynomial with coefficients that are integers. If there is a prime number $\displaystyle b$ that satisfy the following,

$\\a_{n-1}\equiv a_{n-2}\equiv ...\equiv a+0\equiv 0 (\mod b)) \\a_n\not\equiv 0 (\mod b), a_0\not\equiv 0 (\mod b^2)$ $\displaystyle \\a_{n-1}\equiv a_{n-2}\equiv ...\equiv a+0\equiv 0 (\mod b)) \\a_n\not\equiv 0 (\mod b), a_0\not\equiv 0 (\mod b^2)$

Then over the field of rational numbers f(x) $\displaystyle f(x)$ is said to be irreducible.

Report an Error

View Tutors

Dilip
Certified Tutor

University of Calcutta, Bachelor of Science, Physics. Indiana Institute of Technology, Master of Science, Physics.

View Tutors

Patricia
Certified Tutor

Southern Illinois University Edwardsville, Bachelor of Science, Elementary School Teaching. Southern Illinois University Edwa...

View Tutors

Sandra
Certified Tutor

Idaho State University, Bachelor in Arts, Chemistry. Utah State University, Master of Science, Physical Chemistry.

All Abstract Algebra Resources

9 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in Denver, French Tutors in Dallas Fort Worth, Biology Tutors in Phoenix, English Tutors in Houston, SSAT Tutors in Seattle, LSAT Tutors in Washington DC, GRE Tutors in Phoenix, Biology Tutors in Chicago, Math Tutors in New York City, English Tutors in New York City

Popular Courses & Classes

ISEE Courses & Classes in Seattle, ACT Courses & Classes in Atlanta, SAT Courses & Classes in Seattle, MCAT Courses & Classes in Boston, MCAT Courses & Classes in Los Angeles, LSAT Courses & Classes in Denver, ISEE Courses & Classes in Philadelphia, ISEE Courses & Classes in Miami, MCAT Courses & Classes in Atlanta, GMAT Courses & Classes in Los Angeles

Popular Test Prep

ACT Test Prep in Seattle, LSAT Test Prep in San Francisco-Bay Area, MCAT Test Prep in Miami, GRE Test Prep in San Francisco-Bay Area, ISEE Test Prep in Miami, MCAT Test Prep in Philadelphia, GRE Test Prep in Denver, LSAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Philadelphia, GMAT Test Prep in Miami

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

Abstract Algebra : Splitting Fields

Study concepts, example questions & explanations for Abstract Algebra

All Abstract Algebra Resources

Example Questions

Example Question #1 : Fields

All Abstract Algebra Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: