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chwala
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Am going through this notes...kindly let me know if there is a mistake on highlighted part. I think it ought to be;
##α^2=5+2\sqrt{6}##
##α^2=5+2\sqrt{6}##
You are right. It is a mistake in the book and should be ##\alpha^2=5+2\sqrt{6}.## It is the correct value on the list at the end again. The procedure itself is correct.chwala said:Summary: See attached
Am going through this notes...kindly let me know if there is a mistake on highlighted part. I think it ought to be;
##α^2=5+2\sqrt{6}##
View attachment 315473
Thanks...let me peruse through...fresh_42 said:You are right. It is a mistake in the book and should be ##\alpha^2=5+2\sqrt{6}.## It is the correct value on the list at the end again. The procedure it self is correct.
Yes, but Eisenstein uses the fact that ##\mathbb{Z}## is a UFD and ##2## is prime. With that, you don't need Eisenstein anymore:WWGD said:The text from Chwala provides a nice, clean proof of the Irrationality of ##\sqrt 2##, though Eisenstein theorem. Per that theorem, ## x^2 -2## has no Rational solution. A nice, handwavy proof.
A minimal polynomial in ring theory is a monic polynomial of the smallest degree that has a given element as its root. It is used to describe the algebraic relationship between an element and its powers in a ring.
The minimal polynomial can be computed by finding the characteristic polynomial of the given element and then factoring it into irreducible polynomials. The minimal polynomial will be the lowest degree irreducible polynomial in this factorization.
Computing the minimal polynomial is important in ring theory because it helps us understand the algebraic properties of elements in a ring. It can also be used to determine if two elements are algebraically related or if one element is a root of another polynomial.
Minimal polynomials and minimal ideals are related in that a minimal polynomial can be used to generate a minimal ideal in a ring. The minimal polynomial will be the generator of the ideal and all other elements in the ideal can be obtained by multiplying the minimal polynomial with other elements in the ring.
Yes, a minimal polynomial can have multiple roots. This is because the minimal polynomial is defined as the monic polynomial of the smallest degree that has a given element as its root. If the given element has multiple roots, then the minimal polynomial will also have those roots.