Welcome to our community

Be a part of something great, join today!

introduction to linear algebra

abs

New member
Apr 19, 2020
4
prove that $2+8{\sqrt{-5}}$ is unit and irreducible or not in $\mathbb Z+\mathbb Z{\sqrt{-5}}$.
 
Last edited by a moderator:

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,796
prove that $2+8{\sqrt{-5}}$ is unit and irreducible or not in $\mathbb Z+\mathbb Z{\sqrt{-5}}$.
Hint: we can write $2+8{\sqrt{-5}}=2(1+4\sqrt{-5})$.
 

abs

New member
Apr 19, 2020
4

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,796

abs

New member
Apr 19, 2020
4
an element alpha belong to k ia called a unit if alpha divisible by 1.
dear it is my question if u not solved it then no problem its ok .if u solved it then give me complete explanation thank u so much
irreducible element:a non zero non unit element alpha belong to k is said to be irreducible if aplha=ab.
either a is unit or b is unit.
i give u both def. of unit and irreducible thank u so much
 
Last edited by a moderator:

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,796
an element alpha belong to k ia called a unit if alpha divisible by 1.
Not quite.
From wiki:

a unit in a ring with identity $R$ is any element $u$ that has an inverse element in the multiplicative monoid of $R$, i.e. an element $v$ such that
$$uv = vu = 1_R,$$
where $1_R$ is the multiplicative identity​

dear it is my question if u not solved it then no problem its ok .if u solved it then give me complete explanation thank u so much
Sorry, we are a math help site.
We do not usually give complete solutions.
Instead we give hints or similar to help people to learn math.

irreducible element:a non zero non unit element alpha belong to k is said to be irreducible if aplha=ab.
either a is unit or b is unit.
If you're up to it...

The hint I gave showed that we can split the expression in two factors that we might call $a$ and $b$.
Let's start with $2$.
Is it a unit? That is, does it have a multiplicative inverse in the given ring?