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It is well known that $37$ is the first irregular prime, i.e. a prime number $p$ which divides the class number $h_K$ of $K = \mathbb Q(\zeta_{p})$.

For $p=37$, the Hilbert class field $L$ of $K$ is an abelian unramified extension of $K$ with degree $h_K = 37$, hence by Kummer's theory, we have $L = K(\sqrt[p]{\alpha})$, where $\alpha = a + b \zeta_p$ with $a,b \in \mathbb Z$.

I am wondering can you provide such an explicit value of $\alpha = a + b \zeta_p$? More such values for other irregular primes would be very welcome too.

Thanks to the comment of @JoeSilverman, the $\alpha \in \mathbb Z[\zeta_p]$ should be of more general form rather than just in $\mathbb Z + \mathbb Z\cdot \zeta_p$.

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    $\begingroup$ Why is $\alpha$ of the form $a+b\zeta_p$, instead of the more general form $$a_0+a_1\zeta_p+a_2\zeta_p^2+\cdots+a_{p-2}\zeta_p^{p-2},$$ which uses a full basis for $\mathbb Z[\zeta_p]$ as a $\mathbb Z$-module? $\endgroup$ Commented May 24, 2025 at 18:17
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    $\begingroup$ Here is how to obtain a value of $\alpha$ with Pari/GP: bnf = bnfinit(polcyclo(37,y),1); bnr = bnrinit(bnf,1); bnrclassfield(bnr,,1). The output is too large to copy in this comment. $\endgroup$ Commented May 24, 2025 at 18:54
  • $\begingroup$ @JoeSilverman Thank you, you are right, $\alpha$ should be of this form. $\endgroup$ Commented May 24, 2025 at 18:58
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    $\begingroup$ @Aurel, re, I have posted the output as an answer, made CW to avoid reputation. If you prefer that I not do so—for example, if you wanted to post it yourself, or perhaps add context that I did not—then please let me know and I will delete my answer. $\endgroup$ Commented May 24, 2025 at 19:24
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    $\begingroup$ @LSpice Thanks, it's fine this way. $\endgroup$ Commented May 24, 2025 at 20:26

1 Answer 1

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Below is the output of the Pari/GP command bnf = bnfinit(polcyclo(37,y),1); bnr = bnrinit(bnf,1); bnrclassfield(bnr,,1) suggested by @Aurel. This adds nothing to their comment except the bare output, so I have made it CW to avoid reputation.

As one line for convenient C&P:

x^37 + (-6913335969906051693075677*y^35 - 18214360160024149550075633*y^34 - 33577966539673531748444447*y^33 - 52562177950797846151677672*y^32 - 74620838155940362691580735*y^31 - 99119379510197059561693343*y^30 - 125353006370306055312811215*y^29 - 152567032224152455439863668*y^28 - 179978565585983848043283198*y^27 - 206799008757105335444110196*y^26 - 232256808099652247442899768*y^25 - 255619572373874547510069103*y^24 - 276215202997833748305610920*y^23 - 293451207617270477831994744*y^22 - 306831723018020618484333071*y^21 - 315971833260017196672793346*y^20 - 320608579864190573842449755*y^19 - 320608579864190573842449755*y^18 - 315971833260017196672793346*y^17 - 306831723018020618484333071*y^16 - 293451207617270477831994744*y^15 - 276215202997833748305610920*y^14 - 255619572373874547510069103*y^13 - 232256808099652247442899768*y^12 - 206799008757105335444110196*y^11 - 179978565585983848043283198*y^10 - 152567032224152455439863668*y^9 - 125353006370306055312811215*y^8 - 99119379510197059561693343*y^7 - 74620838155940362691580735*y^6 - 52562177950797846151677672*y^5 - 33577966539673531748444447*y^4 - 18214360160024149550075633*y^3 - 6913335969906051693075677*y^2 + 2326752506394055568346493)

and as multiple lines if you prefer to admire it without lots of scrolling:

x^37 + (-6913335969906051693075677*y^35 - 
18214360160024149550075633*y^34 - 33577966539673531748444447*y^33 - 
52562177950797846151677672*y^32 - 74620838155940362691580735*y^31 - 
99119379510197059561693343*y^30 - 125353006370306055312811215*y^29 - 
152567032224152455439863668*y^28 - 179978565585983848043283198*y^27 - 
206799008757105335444110196*y^26 - 232256808099652247442899768*y^25 - 
255619572373874547510069103*y^24 - 276215202997833748305610920*y^23 - 
293451207617270477831994744*y^22 - 306831723018020618484333071*y^21 - 
315971833260017196672793346*y^20 - 320608579864190573842449755*y^19 - 
320608579864190573842449755*y^18 - 315971833260017196672793346*y^17 - 
306831723018020618484333071*y^16 - 293451207617270477831994744*y^15 - 
276215202997833748305610920*y^14 - 255619572373874547510069103*y^13 - 
232256808099652247442899768*y^12 - 206799008757105335444110196*y^11 - 
179978565585983848043283198*y^10 - 152567032224152455439863668*y^9 - 
125353006370306055312811215*y^8 - 99119379510197059561693343*y^7 - 
74620838155940362691580735*y^6 - 52562177950797846151677672*y^5 - 
33577966539673531748444447*y^4 - 18214360160024149550075633*y^3 - 
6913335969906051693075677*y^2 + 2326752506394055568346493)
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