Articles containing proofs | Polynomials | Field (mathematics)

Eisenstein's criterion

In mathematics, Eisenstein's criterion gives a sufficient condition for a polynomial with integer coefficients to be irreducible over the rational numbers – that is, for it to not be factorizable into the product of non-constant polynomials with rational coefficients. This criterion is not applicable to all polynomials with integer coefficients that are irreducible over the rational numbers, but it does allow in certain important cases for irreducibility to be proved with very little effort. It may apply either directly or after transformation of the original polynomial. This criterion is named after Gotthold Eisenstein. In the early 20th century, it was also known as the Schönemann–Eisenstein theorem because Theodor Schönemann was the first to publish it. (Wikipedia).

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Apply the EVT to the square function

👉 Learn how to find the extreme values of a function using the extreme value theorem. The extreme values of a function are the points/intervals where the graph is decreasing, increasing, or has an inflection point. A theorem which guarantees the existence of the maximum and minimum points

From playlist Extreme Value Theorem of Functions

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Does the EVT apply

👉 Learn how to find the extreme values of a function using the extreme value theorem. The extreme values of a function are the points/intervals where the graph is decreasing, increasing, or has an inflection point. A theorem which guarantees the existence of the maximum and minimum points

From playlist Extreme Value Theorem of Functions

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Determine the extrema using EVT of a rational function

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From playlist Extreme Value Theorem of Functions

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Apply the evt and find extrema on a closed interval

👉 Learn how to find the extreme values of a function using the extreme value theorem. The extreme values of a function are the points/intervals where the graph is decreasing, increasing, or has an inflection point. A theorem which guarantees the existence of the maximum and minimum points

From playlist Extreme Value Theorem of Functions

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How to find the extrema using the EVT

👉 Learn how to find the extreme values of a function using the extreme value theorem. The extreme values of a function are the points/intervals where the graph is decreasing, increasing, or has an inflection point. A theorem which guarantees the existence of the maximum and minimum points

From playlist Extreme Value Theorem of Functions

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Determine the extrema of a function on a closed interval

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From playlist Extreme Value Theorem of Functions

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How to apply the evt to a cube root function along a closed interval

👉 Learn how to find the extreme values of a function using the extreme value theorem. The extreme values of a function are the points/intervals where the graph is decreasing, increasing, or has an inflection point. A theorem which guarantees the existence of the maximum and minimum points

From playlist Extreme Value Theorem of Functions

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What is the max and min of a horizontal line on a closed interval

👉 Learn how to find the extreme values of a function using the extreme value theorem. The extreme values of a function are the points/intervals where the graph is decreasing, increasing, or has an inflection point. A theorem which guarantees the existence of the maximum and minimum points

From playlist Extreme Value Theorem of Functions

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Determine the extrema using the end points of a closed interval

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From playlist Extreme Value Theorem of Functions

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Irreducibility (Eisenstein's Irreducibility Criterion)

Given a polynomial with integer coefficients, we can determine whether it's irreducible over the rationals using Eisenstein's Irreducibility Criterion. Unlike some our other technique, this works for polynomials of high degree! The tradeoff is that it works over the rationals, but need not

From playlist Modern Algebra - Chapter 11

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From playlist Abstract Algebra

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RNT2.6.2. Eisenstein's Criterion

Ring Theory: Continuing with Gauss' Lemma, we prove Eisenstein's Criterion for Irreducibility and that R UFD implies R[x] UFD. As an example of EC, we show that f(x) = x^4+x^3+x^2+x+1 is irreducible over the integers using substitution.

From playlist Abstract Algebra

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Abstract Algebra | Eisenstein's criterion

We present a proof of Eisenstein's criterion along with some examples. 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: htt

From playlist Abstract Algebra

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Rings 16 Factorization of polynomials

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From playlist Rings and modules

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How to Prove a Polynomial is Irreducible using Eisenstein's Criterion

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From playlist Abstract Algebra

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CTNT 2022 - The unbounded denominators conjecture (by Yunqing Tang)

This video is one of the special guess talks or conference talks that took place during CTNT 2022, the Connecticut Summer School and Conference in Number Theory. Note: not every special guest lecture or conference lecture was recorded. More about CTNT: https://ctnt-summer.math.uconn.edu/

From playlist CTNT 2022 - Conference lectures and special guest lectures

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This video covers material from Chapter 12 in Robert Redfield's "Abstract Algebra: a Concrete Introduction"

From playlist Modern Algebra - Chapter 12

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Using critical values and endpoints to determine the extrema of a polynomial

👉 Learn how to find the extreme values of a function using the extreme value theorem. The extreme values of a function are the points/intervals where the graph is decreasing, increasing, or has an inflection point. A theorem which guarantees the existence of the maximum and minimum points

From playlist Extreme Value Theorem of Functions

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Irreducible Polynomials

In this video I discuss irreducible polynomials and tests for irreducibility. Note that this video is intended for students in abstract algebra and is not appropriate for high-school or early college level algebra courses.

From playlist Abstract Algebra

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Field of fractions | Perron's irreducibility criterion | Prime ideal | Disquisitiones Arithmeticae | Integral domain | Linear algebra | Frobenius endomorphism | Automorphism | Cohn's irreducibility criterion | Polynomial | Greatest common divisor | Rational number | Theodor Schönemann | Polynomial ring | Lower convex envelope | Lemniscate of Bernoulli | Gaussian integer | Characteristic (algebra) | Binomial coefficient | Mathematics | Integer | Ramification (mathematics) | Gotthold Eisenstein | Algebraic number theory | Leopold Kronecker | Prime number | Cyclotomic polynomial | Irreducible polynomial | Slope | P-adic number | Unique factorization domain | Newton polygon