The polynomial ring, K[X], in X over a fieldK is defined as the set of expressions, called polynomials in X, of the form
where p0, p1,…, pm, the coefficients of p, are elements of K, and X, X2, are formal symbols ("the powers of X"). By convention, X0 = 1, X1 = X, and the product of the powers of X is defined by the familiar formula
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras. The term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra.
Universal algebra is a related subject that studies the nature and theories of various types of algebraic structures as a whole. For example, universal algebra studies the overall theory of groups, as distinguished from studying particular groups.
History
As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra. Through the end of the nineteenth century, many -- perhaps most -- of these problems were in some way related to the theory of algebraic equations. Major themes include:
Abstract Algebra 14.5: Introduction to Polynomial Rings
We have seen that the set of polynomials with real coefficients is a ring, as is the set of polynomials with integer coefficients. In fact, if we start with any commutative ring, we can construct a ring of polynomials where the coefficients come from that ring.
published: 18 Nov 2018
Algebraic Geometry #2: Polynomial Rings
In this video, we introduce the concept of a polynomial ring (its definition, and the notation used). This video doesn't go into some of the nicer properties, e.g. C being isomorphic to the 'quotient-by-ideal' R/(X^2 + 1), but we'll come to that later. For now, I just want to introduce some notation.
---
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published: 22 Jan 2019
Abstract Algebra | Polynomial Rings
We introduce the notion of a polynomial ring, give some examples, and prove a few classic results. In particular we prove that if R is an integral domain then R[x] is as well.
http://www.michael-penn.net
https://www.researchgate.net/profile/Michael_Penn5
http://www.randolphcollege.edu/mathematics/
published: 14 Apr 2020
Polynomial rings 1
Lecture 4
To access the translated content:
1. The translated content of this course is available in regional languages. For details please visit https://nptel.ac.in/translation
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Your feedback is highly appreciated. Kindly fill this form https://forms.gle/XFZhSnHsCLML2LXA6
2. Regional language subtitles available for this course
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published: 23 Aug 2019
Euclidean ring and Polynomial ring | Abstract algebra | Tamil | Limit breaking tamizhaz
Welcome to Limit breaking tamizhaz channel.
Tutor : T.RASIKA
Subject : Abstract Algebra
Topic : Euclidean ring & Polynomial ring
Contents:
Euclidean ring
Polynomial ring
Zero polynomial
Degree of a polynomial
Kindly share your comments and suggestions.
For contact : [email protected]
#limitbreakingtamizhaz
#algebra
#ring
#competitiveexams
#maths_in_tamil
#competitive_exams_preparation
#realanalysis_results
#CSIR_TAMIL
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published: 29 Mar 2021
Polynomial Ring - Introduction - Euclidean Domain - Lesson 11
Download notes from Here:
https://drive.google.com/file/d/1pa2L6JX7M1pDxIzKZ4ICSBYYEZweBKqn/view?usp=sharing
Here in this video i will give the Introduction of Polynomial Ring . From this video i will start the concept of Polynomial Ring in third section of Ring Theory , which is Euclidean Domain.
everything is explained in Hindi
welcome you all in my channel LEARN MATH EASILY
This video will be very useful if you are student of Higher Classes in mathematics like B.Sc, M.Sc , Engineering and if you are preparing for UGC Net and iit Jam etc.
Please Do not forget to Like, Share and Subscribe
Before this topic i did various other topics of Real Analysis:
My other Videos are as follows:
Metric Space
https://www.youtube.com/playlist?list=PLrv011ZmIsBfCoM01KoqDPIYBS7NLFE4P
Countable a...
published: 02 Aug 2020
Abstract Algebra 13.4: A Polynomial Factor Ring
In this video, we consider a more complicated example of a factor ring, and show how it is effectively the same as the complex numbers.
published: 10 Nov 2018
Polynomial Rings and Division -- Abstract Algebra 22
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published: 02 May 2023
Can a Polynomial Have Two Splitting Fields over ℚ???
The definition of a splitting field of a polynomial over a field F is given and the example f(x) = x^2+1 ∈ ℚ[x] over the field of rational numbers ℚ is considered. While this polynomial splits over ℂ, a splitting field is ℚ(i)=ℚ(i,-i) (also, why are these last two fields equal?). Another splitting field is the factor ring ℚ[x]/<x^2+1>. This is a field because x^2+1 is irreducible over ℚ. These two splitting fields are not the same fields, but they are isomorphic to each other. In fact, any two splitting fields of a polynomial f(x) over a field F are isomorphic as fields. So, splitting fields are unique up to isomorphism.
#AbstractAlgebra #FieldTheory #FieldExtension #SplittingField
Links and resources
===============================
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published: 26 Jul 2024
Abstract Algebra | Constructing a field of order 4.
We use the standard strategy involving a quotient of the polynomial ring Z2[x] by a maximal ideal in order to construct a field of order 4.
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We have seen that the set of polynomials with real coefficients is a ring, as is the set of polynomials with integer coefficients. In fact, if we start with an...
We have seen that the set of polynomials with real coefficients is a ring, as is the set of polynomials with integer coefficients. In fact, if we start with any commutative ring, we can construct a ring of polynomials where the coefficients come from that ring.
We have seen that the set of polynomials with real coefficients is a ring, as is the set of polynomials with integer coefficients. In fact, if we start with any commutative ring, we can construct a ring of polynomials where the coefficients come from that ring.
In this video, we introduce the concept of a polynomial ring (its definition, and the notation used). This video doesn't go into some of the nicer properties, e...
In this video, we introduce the concept of a polynomial ring (its definition, and the notation used). This video doesn't go into some of the nicer properties, e.g. C being isomorphic to the 'quotient-by-ideal' R/(X^2 + 1), but we'll come to that later. For now, I just want to introduce some notation.
---
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Video URL: https://youtu.be/J_eV53rHqDY
Channel URL: https://www.youtube.com/channel/UCtTFrEPIqaKfWyGkBi9Rmvg
In this video, we introduce the concept of a polynomial ring (its definition, and the notation used). This video doesn't go into some of the nicer properties, e.g. C being isomorphic to the 'quotient-by-ideal' R/(X^2 + 1), but we'll come to that later. For now, I just want to introduce some notation.
---
SUPPORT ME ON PATREON! Click here: https://www.patreon.com/crystalmath
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Video URL: https://youtu.be/J_eV53rHqDY
Channel URL: https://www.youtube.com/channel/UCtTFrEPIqaKfWyGkBi9Rmvg
We introduce the notion of a polynomial ring, give some examples, and prove a few classic results. In particular we prove that if R is an integral domain then R...
We introduce the notion of a polynomial ring, give some examples, and prove a few classic results. In particular we prove that if R is an integral domain then R[x] is as well.
http://www.michael-penn.net
https://www.researchgate.net/profile/Michael_Penn5
http://www.randolphcollege.edu/mathematics/
We introduce the notion of a polynomial ring, give some examples, and prove a few classic results. In particular we prove that if R is an integral domain then R[x] is as well.
http://www.michael-penn.net
https://www.researchgate.net/profile/Michael_Penn5
http://www.randolphcollege.edu/mathematics/
Lecture 4
To access the translated content:
1. The translated content of this course is available in regional languages. For details please visit https://np...
Lecture 4
To access the translated content:
1. The translated content of this course is available in regional languages. For details please visit https://nptel.ac.in/translation
The video course content can be accessed in the form of regional language text transcripts, books which can be accessed under downloads of each course, subtitles in the video and Video Text Track below the video.
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2. Play the video.
3. Now click on the Settings icon and a list of features will display
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Lecture 4
To access the translated content:
1. The translated content of this course is available in regional languages. For details please visit https://nptel.ac.in/translation
The video course content can be accessed in the form of regional language text transcripts, books which can be accessed under downloads of each course, subtitles in the video and Video Text Track below the video.
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1. Click on the lecture under Course Details.
2. Play the video.
3. Now click on the Settings icon and a list of features will display
4. From that select the option Subtitles/CC.
5. Now select the Language from the available languages to read the subtitle in the regional language.
Welcome to Limit breaking tamizhaz channel.
Tutor : T.RASIKA
Subject : Abstract Algebra
Topic : Euclidean ring & Polynomial ring
Contents:
Euclidean ring
Po...
Welcome to Limit breaking tamizhaz channel.
Tutor : T.RASIKA
Subject : Abstract Algebra
Topic : Euclidean ring & Polynomial ring
Contents:
Euclidean ring
Polynomial ring
Zero polynomial
Degree of a polynomial
Kindly share your comments and suggestions.
For contact : [email protected]
#limitbreakingtamizhaz
#algebra
#ring
#competitiveexams
#maths_in_tamil
#competitive_exams_preparation
#realanalysis_results
#CSIR_TAMIL
#NET_TAMIL
#SET_TAMIL
#TRB_TAMIL
#google
Welcome to Limit breaking tamizhaz channel.
Tutor : T.RASIKA
Subject : Abstract Algebra
Topic : Euclidean ring & Polynomial ring
Contents:
Euclidean ring
Polynomial ring
Zero polynomial
Degree of a polynomial
Kindly share your comments and suggestions.
For contact : [email protected]
#limitbreakingtamizhaz
#algebra
#ring
#competitiveexams
#maths_in_tamil
#competitive_exams_preparation
#realanalysis_results
#CSIR_TAMIL
#NET_TAMIL
#SET_TAMIL
#TRB_TAMIL
#google
Download notes from Here:
https://drive.google.com/file/d/1pa2L6JX7M1pDxIzKZ4ICSBYYEZweBKqn/view?usp=sharing
Here in this video i will give the Introduction of...
Download notes from Here:
https://drive.google.com/file/d/1pa2L6JX7M1pDxIzKZ4ICSBYYEZweBKqn/view?usp=sharing
Here in this video i will give the Introduction of Polynomial Ring . From this video i will start the concept of Polynomial Ring in third section of Ring Theory , which is Euclidean Domain.
everything is explained in Hindi
welcome you all in my channel LEARN MATH EASILY
This video will be very useful if you are student of Higher Classes in mathematics like B.Sc, M.Sc , Engineering and if you are preparing for UGC Net and iit Jam etc.
Please Do not forget to Like, Share and Subscribe
Before this topic i did various other topics of Real Analysis:
My other Videos are as follows:
Metric Space
https://www.youtube.com/playlist?list=PLrv011ZmIsBfCoM01KoqDPIYBS7NLFE4P
Countable and Uncountable Sets
https://www.youtube.com/playlist?list=PLrv011ZmIsBdyWEIOag6HEfyPoq1eIKd7
Supremum & Infimum
https://www.youtube.com/playlist?list=PLrv011ZmIsBdmVB5s1tJSCd39p2rxpigF
Connectedness - Real Analysis
https://www.youtube.com/playlist?list=PLrv011ZmIsBfBXpQkGTeE0SbYoSh0Z0OL
Compactness
https://www.youtube.com/playlist?list=PLrv011ZmIsBdtp8KFmmv6hD2tDUZZbR8a
Neighbourhoods and Limit Points- Real Analysis
https://www.youtube.com/playlist?list=PLrv011ZmIsBcBBN13x8Of5wXlmj1ENowk
Infinite Sequences - Real analysis
https://www.youtube.com/playlist?list=PLrv011ZmIsBcVsSEmckC6_TETuS_4ul-E
Indeterminate forms and l’hospital’s rule
https://www.youtube.com/playlist?list=PLrv011ZmIsBctDlihrQ1dcZKrldqeU-VB
Multiplication Tables- Shortcut tricks
https://www.youtube.com/playlist?list=PLrv011ZmIsBfzdSa7uGjGlilIypG_4gvE
Shortcut tricks to Solve linear equations
https://www.youtube.com/playlist?list=PLrv011ZmIsBeW8FvgL9_TDOBu_2RgTUg9
Quadratic Equations
https://www.youtube.com/playlist?list=PLrv011ZmIsBda4NmnoKPUAHjVRaM1Fn7a
Square and Cube Shortcuts
https://www.youtube.com/playlist?list=PLrv011ZmIsBebZ6V-MA26bAF5vSrvvl1T
Number System
https://www.youtube.com/playlist?list=PLrv011ZmIsBe-TUwGkKf59Bksj_G0eB0L
HCF And LCM
https://www.youtube.com/playlist?list=PLrv011ZmIsBdQchp5I6tw2iSvcPt__szm
Multiplication Tricks
https://www.youtube.com/playlist?list=PLrv011ZmIsBe0StFYtGISxgd6iMt-Yii5
Subscribe to My YouTube Channel " Learn Math Easily" :
https://www.youtube.com/channel/UCUosUwOLsanIozMH9eh95pA
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My Page on Facebook:
https://www.facebook.com/learnmathematicseasily/
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https://youtu.be/TbsLHRadJuEcyclic vbv
Download notes from Here:
https://drive.google.com/file/d/1pa2L6JX7M1pDxIzKZ4ICSBYYEZweBKqn/view?usp=sharing
Here in this video i will give the Introduction of Polynomial Ring . From this video i will start the concept of Polynomial Ring in third section of Ring Theory , which is Euclidean Domain.
everything is explained in Hindi
welcome you all in my channel LEARN MATH EASILY
This video will be very useful if you are student of Higher Classes in mathematics like B.Sc, M.Sc , Engineering and if you are preparing for UGC Net and iit Jam etc.
Please Do not forget to Like, Share and Subscribe
Before this topic i did various other topics of Real Analysis:
My other Videos are as follows:
Metric Space
https://www.youtube.com/playlist?list=PLrv011ZmIsBfCoM01KoqDPIYBS7NLFE4P
Countable and Uncountable Sets
https://www.youtube.com/playlist?list=PLrv011ZmIsBdyWEIOag6HEfyPoq1eIKd7
Supremum & Infimum
https://www.youtube.com/playlist?list=PLrv011ZmIsBdmVB5s1tJSCd39p2rxpigF
Connectedness - Real Analysis
https://www.youtube.com/playlist?list=PLrv011ZmIsBfBXpQkGTeE0SbYoSh0Z0OL
Compactness
https://www.youtube.com/playlist?list=PLrv011ZmIsBdtp8KFmmv6hD2tDUZZbR8a
Neighbourhoods and Limit Points- Real Analysis
https://www.youtube.com/playlist?list=PLrv011ZmIsBcBBN13x8Of5wXlmj1ENowk
Infinite Sequences - Real analysis
https://www.youtube.com/playlist?list=PLrv011ZmIsBcVsSEmckC6_TETuS_4ul-E
Indeterminate forms and l’hospital’s rule
https://www.youtube.com/playlist?list=PLrv011ZmIsBctDlihrQ1dcZKrldqeU-VB
Multiplication Tables- Shortcut tricks
https://www.youtube.com/playlist?list=PLrv011ZmIsBfzdSa7uGjGlilIypG_4gvE
Shortcut tricks to Solve linear equations
https://www.youtube.com/playlist?list=PLrv011ZmIsBeW8FvgL9_TDOBu_2RgTUg9
Quadratic Equations
https://www.youtube.com/playlist?list=PLrv011ZmIsBda4NmnoKPUAHjVRaM1Fn7a
Square and Cube Shortcuts
https://www.youtube.com/playlist?list=PLrv011ZmIsBebZ6V-MA26bAF5vSrvvl1T
Number System
https://www.youtube.com/playlist?list=PLrv011ZmIsBe-TUwGkKf59Bksj_G0eB0L
HCF And LCM
https://www.youtube.com/playlist?list=PLrv011ZmIsBdQchp5I6tw2iSvcPt__szm
Multiplication Tricks
https://www.youtube.com/playlist?list=PLrv011ZmIsBe0StFYtGISxgd6iMt-Yii5
Subscribe to My YouTube Channel " Learn Math Easily" :
https://www.youtube.com/channel/UCUosUwOLsanIozMH9eh95pA
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My Page on Facebook:
https://www.facebook.com/learnmathematicseasily/
.......................................................
https://youtu.be/TbsLHRadJuEcyclic vbv
The definition of a splitting field of a polynomial over a field F is given and the example f(x) = x^2+1 ∈ ℚ[x] over the field of rational numbers ℚ is consider...
The definition of a splitting field of a polynomial over a field F is given and the example f(x) = x^2+1 ∈ ℚ[x] over the field of rational numbers ℚ is considered. While this polynomial splits over ℂ, a splitting field is ℚ(i)=ℚ(i,-i) (also, why are these last two fields equal?). Another splitting field is the factor ring ℚ[x]/<x^2+1>. This is a field because x^2+1 is irreducible over ℚ. These two splitting fields are not the same fields, but they are isomorphic to each other. In fact, any two splitting fields of a polynomial f(x) over a field F are isomorphic as fields. So, splitting fields are unique up to isomorphism.
#AbstractAlgebra #FieldTheory #FieldExtension #SplittingField
Links and resources
===============================
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AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.
The definition of a splitting field of a polynomial over a field F is given and the example f(x) = x^2+1 ∈ ℚ[x] over the field of rational numbers ℚ is considered. While this polynomial splits over ℂ, a splitting field is ℚ(i)=ℚ(i,-i) (also, why are these last two fields equal?). Another splitting field is the factor ring ℚ[x]/<x^2+1>. This is a field because x^2+1 is irreducible over ℚ. These two splitting fields are not the same fields, but they are isomorphic to each other. In fact, any two splitting fields of a polynomial f(x) over a field F are isomorphic as fields. So, splitting fields are unique up to isomorphism.
#AbstractAlgebra #FieldTheory #FieldExtension #SplittingField
Links and resources
===============================
🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinneymath?sub_confirmation=1
🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/
🔴 Follow me on Twitter: https://twitter.com/billkinneymath
🔴 Follow me on Instagram: https://www.instagram.com/billkinneymath/
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🔴 Desiring God website: https://www.desiringgod.org/
AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.
We use the standard strategy involving a quotient of the polynomial ring Z2[x] by a maximal ideal in order to construct a field of order 4.
Please Subscribe: h...
We use the standard strategy involving a quotient of the polynomial ring Z2[x] by a maximal ideal in order to construct a field of order 4.
Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1
Personal Website: http://www.michael-penn.net
Randolph College Math: http://www.randolphcollege.edu/mathematics/
Research Gate profile: https://www.researchgate.net/profile/Michael_Penn5
Google Scholar profile: https://scholar.google.com/citations?user=W5wkSxcAAAAJ&hl=en
We use the standard strategy involving a quotient of the polynomial ring Z2[x] by a maximal ideal in order to construct a field of order 4.
Please Subscribe: https://www.youtube.com/michaelpennmath?sub_confirmation=1
Personal Website: http://www.michael-penn.net
Randolph College Math: http://www.randolphcollege.edu/mathematics/
Research Gate profile: https://www.researchgate.net/profile/Michael_Penn5
Google Scholar profile: https://scholar.google.com/citations?user=W5wkSxcAAAAJ&hl=en
We have seen that the set of polynomials with real coefficients is a ring, as is the set of polynomials with integer coefficients. In fact, if we start with any commutative ring, we can construct a ring of polynomials where the coefficients come from that ring.
In this video, we introduce the concept of a polynomial ring (its definition, and the notation used). This video doesn't go into some of the nicer properties, e.g. C being isomorphic to the 'quotient-by-ideal' R/(X^2 + 1), but we'll come to that later. For now, I just want to introduce some notation.
---
SUPPORT ME ON PATREON! Click here: https://www.patreon.com/crystalmath
Like us on Facebook: https://www.facebook.com/learnmathsfree
Follow us on Twitter: https://twitter.com/LearnMathsFree
Google+: https://plus.google.com/u/0/b/117033642523061853672/117033642523061853672
Video URL: https://youtu.be/J_eV53rHqDY
Channel URL: https://www.youtube.com/channel/UCtTFrEPIqaKfWyGkBi9Rmvg
We introduce the notion of a polynomial ring, give some examples, and prove a few classic results. In particular we prove that if R is an integral domain then R[x] is as well.
http://www.michael-penn.net
https://www.researchgate.net/profile/Michael_Penn5
http://www.randolphcollege.edu/mathematics/
Lecture 4
To access the translated content:
1. The translated content of this course is available in regional languages. For details please visit https://nptel.ac.in/translation
The video course content can be accessed in the form of regional language text transcripts, books which can be accessed under downloads of each course, subtitles in the video and Video Text Track below the video.
Your feedback is highly appreciated. Kindly fill this form https://forms.gle/XFZhSnHsCLML2LXA6
2. Regional language subtitles available for this course
To watch the subtitles in regional languages:
1. Click on the lecture under Course Details.
2. Play the video.
3. Now click on the Settings icon and a list of features will display
4. From that select the option Subtitles/CC.
5. Now select the Language from the available languages to read the subtitle in the regional language.
Welcome to Limit breaking tamizhaz channel.
Tutor : T.RASIKA
Subject : Abstract Algebra
Topic : Euclidean ring & Polynomial ring
Contents:
Euclidean ring
Polynomial ring
Zero polynomial
Degree of a polynomial
Kindly share your comments and suggestions.
For contact : [email protected]
#limitbreakingtamizhaz
#algebra
#ring
#competitiveexams
#maths_in_tamil
#competitive_exams_preparation
#realanalysis_results
#CSIR_TAMIL
#NET_TAMIL
#SET_TAMIL
#TRB_TAMIL
#google
Download notes from Here:
https://drive.google.com/file/d/1pa2L6JX7M1pDxIzKZ4ICSBYYEZweBKqn/view?usp=sharing
Here in this video i will give the Introduction of Polynomial Ring . From this video i will start the concept of Polynomial Ring in third section of Ring Theory , which is Euclidean Domain.
everything is explained in Hindi
welcome you all in my channel LEARN MATH EASILY
This video will be very useful if you are student of Higher Classes in mathematics like B.Sc, M.Sc , Engineering and if you are preparing for UGC Net and iit Jam etc.
Please Do not forget to Like, Share and Subscribe
Before this topic i did various other topics of Real Analysis:
My other Videos are as follows:
Metric Space
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Neighbourhoods and Limit Points- Real Analysis
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Infinite Sequences - Real analysis
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Indeterminate forms and l’hospital’s rule
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Multiplication Tables- Shortcut tricks
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Shortcut tricks to Solve linear equations
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HCF And LCM
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https://youtu.be/TbsLHRadJuEcyclic vbv
The definition of a splitting field of a polynomial over a field F is given and the example f(x) = x^2+1 ∈ ℚ[x] over the field of rational numbers ℚ is considered. While this polynomial splits over ℂ, a splitting field is ℚ(i)=ℚ(i,-i) (also, why are these last two fields equal?). Another splitting field is the factor ring ℚ[x]/<x^2+1>. This is a field because x^2+1 is irreducible over ℚ. These two splitting fields are not the same fields, but they are isomorphic to each other. In fact, any two splitting fields of a polynomial f(x) over a field F are isomorphic as fields. So, splitting fields are unique up to isomorphism.
#AbstractAlgebra #FieldTheory #FieldExtension #SplittingField
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We use the standard strategy involving a quotient of the polynomial ring Z2[x] by a maximal ideal in order to construct a field of order 4.
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The polynomial ring, K[X], in X over a fieldK is defined as the set of expressions, called polynomials in X, of the form
where p0, p1,…, pm, the coefficients of p, are elements of K, and X, X2, are formal symbols ("the powers of X"). By convention, X0 = 1, X1 = X, and the product of the powers of X is defined by the familiar formula
They apply GSR-NTT to accelerate polynomial multiplication in the lattice-based scheme named NTTRU and single polynomial multiplication over power-of-three cyclotomic polynomial rings.
A revolution in ring theory ...Ring theory is the study of mathematical objects called rings ... A ring can be made of numbers, functions, matrices, polynomials or other abstract objects – as long as there’s a way to add, subtract and multiply them.
A revolution in ring theory ...Ring theory is the study of mathematical objects called rings ... A ring can be made of numbers, functions, matrices, polynomials or other abstract objects – as long as there’s a way to add, subtract and multiply them.