Anki Deck Changes

Note 1: ETH::DiskMat

Deck: ETH::DiskMat
Note Type: Horvath Cloze
GUID: Cx

added

Note did not exist

In a Euclidean domain every element can be factored uniquely into irreducible elements (up to associates)

In a Euclidean domain every element can be factored uniquely into irreducible elements (up to associates)

$a, b$ associates ($a \sim b$) if $a = ub$ for some unit $u$.

Field-by-field Comparison

Field	Before	After
Text		In a Euclidean domain every element can be {{c1:: factored uniquely into irreducible elements (up to associates)}}
Extra		$a, b$ associates ($a \sim b$) if $a = ub$ for some unit $u$.

Tags: ETH::1._Semester::DiskMat::5._Algebra::6._Polynomials_over_a_Field::3._Analogies_Between_Z_and_F[x],_Euclidean_Domains_*

Deck: ETH::DiskMat
Note Type: Horvath Classic
GUID: pa$vtGEA[j

added

Note did not exist

Define a euclidean domain:

Define a euclidean domain:

A euclidean domain is an integral domain $D$ together with a degree function $d: D \setminus {0} \rightarrow \mathbb{N}$ such that:

For every $a$ and $b \neq 0$ in $D$ there exist $q$ and $r$ such that $a = bq + r$ and $d(r) < d(b)$ or $r = 0$
For all nonzero $a$ and $b$ in $D$, $d(a) \leq d(ab)$.

Field-by-field Comparison

Field	Before	After
Front		Define a euclidean domain:
Back		<div>A euclidean domain is an integral domain $D$ together with a degree function $d: D \setminus {0} \rightarrow \mathbb{N}$ such that:</div><ul><li>For every $a$ and $b \neq 0$ in $D$ there exist $q$ and $r$ such that $a = bq + r$ and $d(r) < d(b)$ or $r = 0$</li><li>For all nonzero $a$ and $b$ in $D$, $d(a) \leq d(ab)$.</li></ul>

Tags: ETH::1._Semester::DiskMat::5._Algebra::6._Polynomials_over_a_Field::3._Analogies_Between_Z_and_F[x],_Euclidean_Domains_*