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Our current lounge which adjoins the kitchen!

Finding a degree 4 extension of Q adjoin root (-21) that is a PID?

This question seems like it should be easy, but it's destroying me. Discriminant is -84, minkowski bound is 5.8ish so that we only need consider p=2,3,5. Let root(-21)=r, then Kummer factorization gives that

2R=(2, 1+ r)^2,

3R=(3,r)^2

5R=(5,2+r)(5,2-r)

it seems apparent we are supposed to adjoin two of the square roots of 2,3,5, or 7 (or perhaps root -2,-3,-5,-7). correct me if this is wrong. By taking Norms we see any element in our extension generating a principal ideal that divides 2R has norm 2, similarly for 3R has norm 3, and for 5R has norm 5. This is as far as I've seriously gotten: I've tooled around for a while with adjoining pairs and checking for elements of valid norms (which I list below) but nothing has worked. I am thinking I may need to focus more on adjoining negative roots.

Considering real roots, elements of norm 2 are sqrt(2), sqrt(3)-1, sqrt(7)-sqrt(5), sqrt(5)-sqrt(3), and their conjugates (I'm definitely missing some good candidates but these seem reasonable). Elements of norm 3 are sqrt(3), sqrt(5)-sqrt(2), sqrt(7)-2, etc. Elements of norm 5 are sqrt(5), sqrt(7)-sqrt(2), 2sqrt(2)-sqrt(3), etc. Essentially I think that it is not possible to have an element of norm 5 by adjoining both sqrt(3), sqrt(7) but I honestly have no clue.

If some of this makes no sense, I apologize because these computations are proving pretty difficult for me to wrap my head around. Can someone suggest a valid pair along with perhaps some generators? Thank you!

Adjoin Garden / July Cooperative Company archdaily.com/950188/adjoβ¦

Got the keys to my new house which adjoins the golf course Iβm a member of. I havenβt come home before dark on one occasion since settling in! I love it! v.redd.it/aoiub4usl9r51

This beautiful beach adjoins a town with the same name which had major environmental problems through the 1990s.

Iβm getting back into playing for the first time since my senior year of highschool now that Iβm finally recovered from adjoins hockey and life related injuries was wondering how i should go about trying to shoot a puck around again before going to my schools club team.

I also need to pick up new knee pads since my old ones got shattered towards the end of my playing time. Iβm in chicago so i wouldnβt be surprised if the rinks are still under some limited capacity near me.

Anyone know what these are? Found a ton of them where this redwood grass border adjoins to the fence.

My shop adjoins the premium luxury wing of our service building. Sometimes I walk into the lot in the morning to see a casual F40. Apparently Scottsdale and PV is full of F40s just sitting. I've seen at least 6-7 here in the past year and a half for refurbs or services after storage.

Unsound-but-awesome lens combinators: lensProduct and adjoin tek.brick.do/96a83b0f-d2fβ¦

To the kids spray painting the sides of a big boring grey warehouse that adjoins abandoned land

The fog was so thick you couldn't be seen, very clever and I look forward to seeing your art on my next walk

Model Y tailgate *IS* the rear bumper?! Having been rear-ended in my S, this is concerning... especially since the top edge adjoins a very expensive piece of glass.

Adjoin by Nick Sullo

After deciding my new AnyCubic Photon is just too slick looking to banish to the basement, I decided it gets to live in my office with the Ender3 Pro. Since that area adjoins my living room, I got some things to mitigate resin fumes. Printed up a rear duct adapter and the filter just arrived!

TIL Graveyard is a burial ground that adjoins a churchβs and a cemetery is a burial ground that does not. People are buried in graveyards but in cemeteries it can be both people and ashes. thehindu.com/books/know-yβ¦

Deepweed (Adjoin by XSullo)

Adjoin by Nick Sullo

TIL I learned the difference between a graveyard and a cemetery. A graveyard adjoins a church, whereas a cemetery does not. You can also bury ashes in a cemetery. thehindu.com/books/know-yβ¦

[ANCAPS] In an ANCAP utopia how many toll roads will adjoin my house?

I live in rural America. How many companies can I expect to offer toll road services to my house without government getting in the way?

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A little piece of art we crafted today,with help of my mentor managed to adjoin two different piece altogether with crocheting ,beads & stone.We named it βNecChokerβ FINAL RESULT π

You can add a new state off any of the U.S. coasts, but what state do you adjoin it to, and what's its history?

The alternate United States of America is something occasionally used in fiction but what would be your new states [let's assume you could add 10 more states to the U.S.]?

What do you think the state would be like culturally, geographically etc., and in terms of demographics, political voting etc.? What would the food be like?

I'm interested to know.

Jacob's Well is located northwest of Wimberley, Texas. The 12-foot (3.7 m) diameter mouth of the spring serves as a popular swimming spot for the local land owners whose properties adjoin Cypress Creek. en.wikipedia.org/wiki/Jacβ¦

How do I adjoin 4x4's?

I want to build this bed.

How do I go about adjoining the 4x4's? Just wood glue?

[Room] Well lit sitting area adjoins the kitchen in this completely renovated mid-century home located in Waco, Texas. [1000 Γ 667]

TIL The Oval Office Study is the working office of the President of the United States. It adjoins the Oval Office, the ceremonial office of the president and is accessed from a small corridor outside the Oval Office, which includes the president's private lavatory and the president's dining room en.wikipedia.org/wiki/Ovaβ¦

Well lit sitting area adjoins the kitchen in this completely renovated mid-century home located in Waco, Texas. [1000 Γ 667]

LPT: If your area's tennis courts adjoin wooded areas, creep around in the brush and you will find plenty of lost tennis balls, most in good enough shape to use -- a money saver!

Haven't bought any since my kids took up the sport a few years back.

TIL I learned the difference between a graveyard and a cemetery. A graveyard adjoins a church, whereas a cemetery does not. You can also bury ashes in a cemetery. - todayilearned reddit.com/r/todayilearneβ¦

Sigma Labs Inc (OTCMKTS:SGLB) Adjoins Hands With America Makes journaltranscript.com/201β¦

I'm fairly certain that my landlord tried to burn down the building that adjoins my current residence.

Let's call this what it is...."Jewish Lightning". It is a derogatory but very true term that many of you living in urban areas should be familiar with.

I'm kinda freaked out now and won't be staying there tonight. Short form question is, what should be the first legal action I take tomorrow? None of my stuff is destroyed technically, but the house is filled with acrid smoke leftovers and who knows when these fuckfaces might be back. I'm just really pissed off and trying to find something constructive to direct this energy towards. Yea, I live in a place that leaves much to be desired but the price is right, however, this crossed a line.

The firemen and cops seemed to know what's up and were fairly casual about it. I'm a bit more upset, as this extremely intense fire happened about 20ft from where I sleep. Any help, or just telling me to have a beer, a toke, and chill would be appreciated. Thanks.

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