Galois Group of x^4-2

[@ravikantpatil3398]

Abstract algebra is interesting to me

[@terrym2007]

Check if the Galois group of x^4-2 is Z2xZ2, not D8 over Q..Thanks

[@eliwebinger7737]

Came here for this.
I think he meant D4 as D_8 has 16 elements, not 8.
D_4 would be the correct answer. The little sketch with the square was also correct, not being an octagon.
After sylow, there are 3 subgroups of order 8 which are all isomorphic to D4.

[@SabyasachiGhosh1618]

@eliwebinger there are differing naming conventions, sometimes the group of symmetries of an n-gon is called D_(2n) instead of D_n. The book Dummit and Foote follows this convention for example

[@eliwebinger7737]

@@SabyasachiGhosh1618 thanks! I didn't know!

[@coconutmath4928]

@@SabyasachiGhosh1618 It seems like this conversation was resolved already but yes, it seems like there are different naming conventions. I would usually refer to it as D_8 (or D_10 or D_12 or whatever) since that's the number of elements of the group. But as far as I know it's just personal preference

[@AmadeusDeAmicis-e3v]

yes Dn vs D2n naming convention is a EU vs US thing, never remember which is which but clear from context

[@juliefinkjulesheartmagic1111]

Love your videos. They make math more fun!

[@poojasingh7200]

Exactly

[@pradeepbose159]

Sir please upload this : prove that the field extension Q(√5+i) over Q is an algebraic extension

[@coconutmath4928]

Uploaded. Let me know if you have any questions!

[@pradeepbose159]

@@coconutmath4928 thank you sir ❤️ from India

[@tomandjerry7524]

Sir if we want to find fixed field from the all proper subgroups of G then
How can we pricisely write all fixed fields
Is are you teach ?

[@AmadeusDeAmicis-e3v]

1. fixes all roots = identity element e
2. flips i and -i and fixes all the rest
3. flips sqrt(2) and - sqrt(2) and fixes all the rest
4. flips 4th-root of 2 and minus 4th- root of 2 and fixes all the rest
5. flips i/-i and sqrt2/-sqrt2 and fixes all the rest
6. flips i/-i and 4th-root2/-4th-root2 and fixes all the rest
7. flips sqrt2/-srqt2 and 4th-root2/-4th-root2 and fixes all the rest
8. flips i/-i and sqrt2/-srqt2 and 4th-root2/-4th-root2 and fixes all the rest (the rest here is the basefield Q only)
Hope this helps

[@elisabethseguin582]

This is quite good. However, because gcd(4,2) is not 1, who do you conclude that the stacked extensions degree is the product of the individual degrees?

[@coconutmath4928]

Thanks! That result is always true, see en.wikipedia.org/wiki/Degree_of_a_field_extension#The_multiplicativity_formula_for_degrees

[@AmadeusDeAmicis-e3v]

well done

[@omargaber3122]

Great, thank you

[@karansirmathematics]

Which software are you using to make videos

[@coconutmath4928]

I use microsoft whiteboard and record with OBS studio. OBS is free and records via window capture. Whiteboard is not great but gets the job done haha

[@karansirmathematics]

Thanks bro 👍
You are doing great job

[@karansirmathematics]

Great

[@coconutmath4928]

Thanks very much!

[@poojasingh7200]

Thanks

[@coconutmath4928]

Of course! Hope the video was helpful.

[@emmaly7596]

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