Cyclotomic Field Puns

Just doing a first course in Galois theory atm. Any insight or links is appreciated :)

Hey everyone! I was doing exercise 14.7.33 from Dummit and Foote and managed to do part (a) but I've been trying to do part (b) with no success. In part (a) I proved that the discriminant of the cyclotomic polynomial Phi_p for an odd prime p is (-1)^ ((p-1) /2) *p^ (p-2). Now part (b) says to prove that (and I'm using Q for \mathhbb{Q}, obviously): Q(sqrt((-1)^ ((p-1)/2) p)) is properly contained in Q(zeta_p), where zeta_p is a primitive pth root of unity. I thought that the fact that the square root of the discriminant of a polynomial is always in the splitting field of the polynomial would be useful, but I tried to use it and nothing came out of it. Can someone please help me? Thanks!

I know what Kummer's criterion is, and I want to know further, for example Iwasawa theory stuff. What do we know about ideal class group of cyclotomic fields?

I have been asked to give a presentation in class discussing the Galois theory of the n-th cyclotomic field integer n (odd or divisible by four) which has at least two prime factors and is between twenty and ninety. Number theory is definitely not my friend so I will have to do a lot of preparation, but instead of picking one at random I was wondering if there is an example with particularly nice features, which I could use to make my presentation more interesting and less just a bunch of boring calculations. Any help or pointers would be much appreciated!

In another thread, I made a slightly satirical comment that the "fundamental constant" 2𝜋i already has a name in mathematics (called "(1)") and got downvoted into oblivion. I figured that perhaps I should make a totally un-satirical post justifying why this is the case, so what follows will be a little deeper than the 𝜋-vs.-𝜏 debate. (Edit: I'm not trying to vindicate myself- I just want to explain some ideas behind this notation due to Deligne, which I find beautiful.)

To be absolutely precise, it is not the constant 2𝜋i that occupies a fundamental role in mathematics, but rather the abelian group (2𝜋i)Z. The latter is written as Z(1) and is pronounced the "Tate twist of Z". A better definition of Z(1)--one that is visibly independent of the choice of a square root of (-1)--is that it is dual to the second integral cohomology of the complex projective line P1. Why does this definition recover (2𝜋i)Z? Well, you may know that the Picard group of P1 is canonically isomorphic to Z. On the other hand, the exponential exact sequence tells us that this Picard group is isomorphic to H^2 with coefficients in the kernel of the exponential map. If a square root of (-1) is fixed (called i), then this kernel is generated by 2𝜋i. If you wish, this is actually the definition of 𝜋 via circumference of the circle. (Related fact: the first Chern class of a holomorphic line bundle lives in (2𝜋i)H^2(X; Z) rather than H^2(X; Z).)

If you come from an algebraic background, you may have already seen Tate twists but in the context of Galois modules. Let me convince you that the 1-dimensional Galois module known as the "Tate twist" is again a manifestation of the second cohomology of the projective line. Working over an arbitrary field k, we may replace topological cohomology by étale cohomology with Z/n coefficients (for an integer n invertible in k). It turns out that the dual of the second étale cohomology along P1_k -> Spec(k) is not Z/n but rather 𝜇_n, the étale sheaf of (n)th roots of unity (Kummer theory). Translating between étale sheaves over Spec(k) and modules over the absolute Galois group of k, you see that 𝜇_n corresponds to the cyclotomic character (i.e., the "Pick a primitive Nth root of unity and read off the exponent" character). This is why the étale sheaf 𝜇_n is the Tate twist of Z/n, and is often denoted by Z/n(1). Historically, this came before the notation in the complex case.

The idea that these two drastically different worlds are

I don't want to step on anybody's toes here, but the amount of non-dad jokes here in this subreddit really annoys me. First of all, dad jokes CAN be NSFW, it clearly says so in the sub rules. Secondly, it doesn't automatically make it a dad joke if it's from a conversation between you and your child. Most importantly, the jokes that your CHILDREN tell YOU are not dad jokes. The point of a dad joke is that it's so cheesy only a dad who's trying to be funny would make such a joke. That's it. They are stupid plays on words, lame puns and so on. There has to be a clever pun or wordplay for it to be considered a dad joke.

Again, to all the fellow dads, I apologise if I'm sounding too harsh. But I just needed to get it off my chest.

Do your worst!

I'm surprised it hasn't decade.

For context I'm a Refuse Driver (Garbage man) & today I was on food waste. After I'd tipped I was checking the wagon for any defects when I spotted a lone pea balanced on the lifts.

I said "hey look, an escaPEA"

No one near me but it didn't half make me laugh for a good hour or so!

Edit: I can't believe how much this has blown up. Thank you everyone I've had a blast reading through the replies 😂

It really does, I swear!

They’re on standbi

Pilot on me!!

Nothing, he was gladiator.

Dad jokes are supposed to be jokes you can tell a kid and they will understand it and find it funny.

This sub is mostly just NSFW puns now.

If it needs a NSFW tag it's not a dad joke. There should just be a NSFW puns subreddit for that.

Edit* I'm not replying any longer and turning off notifications but to all those that say "no one cares", there sure are a lot of you arguing about it. Maybe I'm wrong but you people don't need to be rude about it. If you really don't care, don't comment.

When I got home, they were still there.

What did 0 say to 8 ?

" Nice Belt "

So What did 3 say to 8 ?

" Hey, you two stop making out "

I won't be doing that today!

[Removed]

Where ever you left it 🤷‍♀️🤭

This morning, my 4 year old daughter.

Daughter: I'm hungry

Me: nerves building, smile widening

Me: Hi hungry, I'm dad.

She had no idea what was going on but I finally did it.

Thank you all for listening.

You take away their little brooms

There hasn't been a post all year!

It was about a weak back.

Because she wanted to see the task manager.

Why

It’s pronounced “Noel.”

After all his first name is No-vac

What, then, is Chinese rap?

Edit:

Notable mentions from the comments:

• Spanish/Swedish/Swiss/Serbian hits

• French/Finnish art

• Country/Canadian rap

• Chinese/Country/Canadian rock

• Turkish/Tunisian/Taiwanese rap

There hasn't been a single post this year!

(Happy 2022 from New Zealand)

Nothing, it just waved

Him: I can explain everything!

(It's his best joke yet I think)

Bob