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Is there an elementary proof that there are infinitely many primes that are *not* completely split in an abelian extension?


Is there an “elementary” proof of the infinitude of completely split primes?How many primes stay inert in a finite (non-cyclic) extension of number fields?Is there an “elementary” proof of the infinitude of completely split primes?For what subsets S of (Z/nZ)* is there a Euclidean proof that there are infinitely many primes whose residues lie in S?Where does the principal ideal theorem (from CFT) go?Characterizing primes that split completely vs. primes with a given splitting behaviorDensity of primes of degree one in Bauer's Theorem (Application of Chebotarev Density)Elementary Proof of Infinitely many primes $mathfrakp in mathbbZ[i]$ in the sector $theta < arg mathfrakp <phi $Rank growth in ray class fields of primes that are inert in an imaginary quadratic extensionAn elementary proof that, for every fixed $n in mathbf N^+$, there are infinitely many primes $equiv -1 bmod n$Existence of infinitely many Heegner points that are divisible by $p^n$ in $K_lambda$













9












$begingroup$


I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following:



Lemma: if L/K is an abelian extension of number fields, then there are infinitely many primes of K that do not split competely in L.



Of course this is implied by Cebotarev's density theorem, but the adelic proof uses only algebra/topology and finiteness of class number/Dirichlet's units theorem.



There is a well-known elementary proof, (see eg this MO question) that there are infinitely many primes that are split in L/K. I was wondering whether there is also an elementary argument for infinitude of non-split primes in the extension? (As usual the notion of "elementary" is flexible, but I'm looking for something that uses a minimum of machinery.)



One possibility would be to distill the adelic proof into something algebraic, although this seems hard. Another option would be to look for ideals of O_K that are not norms from O_L: any such ideal must have a factor which does not split completely.



One of the answers to the MathOverflow question linked above does mention the paper Primes of degree one and algebraic cases of Čebotarev's theorem of Lenstra and Stevenhagen, which gives an elementary proof under the assumption that L contains a nontrivial ray class field of K. But it seems that one still needs to prove the first inequality in some form to use this.










share|cite|improve this question











$endgroup$







  • 3




    $begingroup$
    If $K=mathbbQ$ this follows from Kronecker-Weber. Suppose $LsubsetmathbbQ(zeta_n)$ corresponds to a proper subgroup $Hleq (mathbbZ/nmathbbZ)^times$. A rational prime $pnmid n$ splits completely in $L$ iff $pmod n$ is in $H$. If $p_1,ldots,p_m$ are primes not dividing $n$ that do not split completely, we can pick $rin(mathbbZ/nmathbbZ)^timesbackslash H$ and use the CRT to find a positive integer $Q$ congruent to $1$ mod $p_1cdots p_m$ and congruent to $r$ mod $n$. Then $Q$ must have a prime divisor that does not split completely, which is not $p_1,ldots,p_m$.
    $endgroup$
    – Julian Rosen
    9 hours ago











  • $begingroup$
    @JulianRosen Yes, and I believe that's the argument that Lenstra and Stevenhagen generalize to arbitrary ray class fields.
    $endgroup$
    – Alison Miller
    9 hours ago















9












$begingroup$


I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following:



Lemma: if L/K is an abelian extension of number fields, then there are infinitely many primes of K that do not split competely in L.



Of course this is implied by Cebotarev's density theorem, but the adelic proof uses only algebra/topology and finiteness of class number/Dirichlet's units theorem.



There is a well-known elementary proof, (see eg this MO question) that there are infinitely many primes that are split in L/K. I was wondering whether there is also an elementary argument for infinitude of non-split primes in the extension? (As usual the notion of "elementary" is flexible, but I'm looking for something that uses a minimum of machinery.)



One possibility would be to distill the adelic proof into something algebraic, although this seems hard. Another option would be to look for ideals of O_K that are not norms from O_L: any such ideal must have a factor which does not split completely.



One of the answers to the MathOverflow question linked above does mention the paper Primes of degree one and algebraic cases of Čebotarev's theorem of Lenstra and Stevenhagen, which gives an elementary proof under the assumption that L contains a nontrivial ray class field of K. But it seems that one still needs to prove the first inequality in some form to use this.










share|cite|improve this question











$endgroup$







  • 3




    $begingroup$
    If $K=mathbbQ$ this follows from Kronecker-Weber. Suppose $LsubsetmathbbQ(zeta_n)$ corresponds to a proper subgroup $Hleq (mathbbZ/nmathbbZ)^times$. A rational prime $pnmid n$ splits completely in $L$ iff $pmod n$ is in $H$. If $p_1,ldots,p_m$ are primes not dividing $n$ that do not split completely, we can pick $rin(mathbbZ/nmathbbZ)^timesbackslash H$ and use the CRT to find a positive integer $Q$ congruent to $1$ mod $p_1cdots p_m$ and congruent to $r$ mod $n$. Then $Q$ must have a prime divisor that does not split completely, which is not $p_1,ldots,p_m$.
    $endgroup$
    – Julian Rosen
    9 hours ago











  • $begingroup$
    @JulianRosen Yes, and I believe that's the argument that Lenstra and Stevenhagen generalize to arbitrary ray class fields.
    $endgroup$
    – Alison Miller
    9 hours ago













9












9








9


1



$begingroup$


I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following:



Lemma: if L/K is an abelian extension of number fields, then there are infinitely many primes of K that do not split competely in L.



Of course this is implied by Cebotarev's density theorem, but the adelic proof uses only algebra/topology and finiteness of class number/Dirichlet's units theorem.



There is a well-known elementary proof, (see eg this MO question) that there are infinitely many primes that are split in L/K. I was wondering whether there is also an elementary argument for infinitude of non-split primes in the extension? (As usual the notion of "elementary" is flexible, but I'm looking for something that uses a minimum of machinery.)



One possibility would be to distill the adelic proof into something algebraic, although this seems hard. Another option would be to look for ideals of O_K that are not norms from O_L: any such ideal must have a factor which does not split completely.



One of the answers to the MathOverflow question linked above does mention the paper Primes of degree one and algebraic cases of Čebotarev's theorem of Lenstra and Stevenhagen, which gives an elementary proof under the assumption that L contains a nontrivial ray class field of K. But it seems that one still needs to prove the first inequality in some form to use this.










share|cite|improve this question











$endgroup$




I'm currently in the middle of teaching the adelic algebraic proofs of global class field theory. One of the intermediate lemmas that one shows is the following:



Lemma: if L/K is an abelian extension of number fields, then there are infinitely many primes of K that do not split competely in L.



Of course this is implied by Cebotarev's density theorem, but the adelic proof uses only algebra/topology and finiteness of class number/Dirichlet's units theorem.



There is a well-known elementary proof, (see eg this MO question) that there are infinitely many primes that are split in L/K. I was wondering whether there is also an elementary argument for infinitude of non-split primes in the extension? (As usual the notion of "elementary" is flexible, but I'm looking for something that uses a minimum of machinery.)



One possibility would be to distill the adelic proof into something algebraic, although this seems hard. Another option would be to look for ideals of O_K that are not norms from O_L: any such ideal must have a factor which does not split completely.



One of the answers to the MathOverflow question linked above does mention the paper Primes of degree one and algebraic cases of Čebotarev's theorem of Lenstra and Stevenhagen, which gives an elementary proof under the assumption that L contains a nontrivial ray class field of K. But it seems that one still needs to prove the first inequality in some form to use this.







nt.number-theory algebraic-number-theory class-field-theory number-fields elementary-proofs






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited 7 hours ago







Alison Miller

















asked 10 hours ago









Alison MillerAlison Miller

3,01312128




3,01312128







  • 3




    $begingroup$
    If $K=mathbbQ$ this follows from Kronecker-Weber. Suppose $LsubsetmathbbQ(zeta_n)$ corresponds to a proper subgroup $Hleq (mathbbZ/nmathbbZ)^times$. A rational prime $pnmid n$ splits completely in $L$ iff $pmod n$ is in $H$. If $p_1,ldots,p_m$ are primes not dividing $n$ that do not split completely, we can pick $rin(mathbbZ/nmathbbZ)^timesbackslash H$ and use the CRT to find a positive integer $Q$ congruent to $1$ mod $p_1cdots p_m$ and congruent to $r$ mod $n$. Then $Q$ must have a prime divisor that does not split completely, which is not $p_1,ldots,p_m$.
    $endgroup$
    – Julian Rosen
    9 hours ago











  • $begingroup$
    @JulianRosen Yes, and I believe that's the argument that Lenstra and Stevenhagen generalize to arbitrary ray class fields.
    $endgroup$
    – Alison Miller
    9 hours ago












  • 3




    $begingroup$
    If $K=mathbbQ$ this follows from Kronecker-Weber. Suppose $LsubsetmathbbQ(zeta_n)$ corresponds to a proper subgroup $Hleq (mathbbZ/nmathbbZ)^times$. A rational prime $pnmid n$ splits completely in $L$ iff $pmod n$ is in $H$. If $p_1,ldots,p_m$ are primes not dividing $n$ that do not split completely, we can pick $rin(mathbbZ/nmathbbZ)^timesbackslash H$ and use the CRT to find a positive integer $Q$ congruent to $1$ mod $p_1cdots p_m$ and congruent to $r$ mod $n$. Then $Q$ must have a prime divisor that does not split completely, which is not $p_1,ldots,p_m$.
    $endgroup$
    – Julian Rosen
    9 hours ago











  • $begingroup$
    @JulianRosen Yes, and I believe that's the argument that Lenstra and Stevenhagen generalize to arbitrary ray class fields.
    $endgroup$
    – Alison Miller
    9 hours ago







3




3




$begingroup$
If $K=mathbbQ$ this follows from Kronecker-Weber. Suppose $LsubsetmathbbQ(zeta_n)$ corresponds to a proper subgroup $Hleq (mathbbZ/nmathbbZ)^times$. A rational prime $pnmid n$ splits completely in $L$ iff $pmod n$ is in $H$. If $p_1,ldots,p_m$ are primes not dividing $n$ that do not split completely, we can pick $rin(mathbbZ/nmathbbZ)^timesbackslash H$ and use the CRT to find a positive integer $Q$ congruent to $1$ mod $p_1cdots p_m$ and congruent to $r$ mod $n$. Then $Q$ must have a prime divisor that does not split completely, which is not $p_1,ldots,p_m$.
$endgroup$
– Julian Rosen
9 hours ago





$begingroup$
If $K=mathbbQ$ this follows from Kronecker-Weber. Suppose $LsubsetmathbbQ(zeta_n)$ corresponds to a proper subgroup $Hleq (mathbbZ/nmathbbZ)^times$. A rational prime $pnmid n$ splits completely in $L$ iff $pmod n$ is in $H$. If $p_1,ldots,p_m$ are primes not dividing $n$ that do not split completely, we can pick $rin(mathbbZ/nmathbbZ)^timesbackslash H$ and use the CRT to find a positive integer $Q$ congruent to $1$ mod $p_1cdots p_m$ and congruent to $r$ mod $n$. Then $Q$ must have a prime divisor that does not split completely, which is not $p_1,ldots,p_m$.
$endgroup$
– Julian Rosen
9 hours ago













$begingroup$
@JulianRosen Yes, and I believe that's the argument that Lenstra and Stevenhagen generalize to arbitrary ray class fields.
$endgroup$
– Alison Miller
9 hours ago




$begingroup$
@JulianRosen Yes, and I believe that's the argument that Lenstra and Stevenhagen generalize to arbitrary ray class fields.
$endgroup$
– Alison Miller
9 hours ago










1 Answer
1






active

oldest

votes


















11












$begingroup$

I mention this as an answer since it is too long for comments. I do not know what the adelic proof assumes. Suppose that all but finitely many primes of $K$ split completely in $L$. Suppose $d$ is the degree of $L$ over $K$. Then the zeta function of $L$ is the $d$-th power of the zeta function of $K$, up to finitely many factors. But the Zeta functions of $L$ and $K$ have only a simple pole at $s=1$ (implied by finiteness of class number ...). Hence $d=1$ and $L=K$.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Good point! That's certainly simpler than the adelic proof.
    $endgroup$
    – Alison Miller
    9 hours ago










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1 Answer
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active

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11












$begingroup$

I mention this as an answer since it is too long for comments. I do not know what the adelic proof assumes. Suppose that all but finitely many primes of $K$ split completely in $L$. Suppose $d$ is the degree of $L$ over $K$. Then the zeta function of $L$ is the $d$-th power of the zeta function of $K$, up to finitely many factors. But the Zeta functions of $L$ and $K$ have only a simple pole at $s=1$ (implied by finiteness of class number ...). Hence $d=1$ and $L=K$.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Good point! That's certainly simpler than the adelic proof.
    $endgroup$
    – Alison Miller
    9 hours ago















11












$begingroup$

I mention this as an answer since it is too long for comments. I do not know what the adelic proof assumes. Suppose that all but finitely many primes of $K$ split completely in $L$. Suppose $d$ is the degree of $L$ over $K$. Then the zeta function of $L$ is the $d$-th power of the zeta function of $K$, up to finitely many factors. But the Zeta functions of $L$ and $K$ have only a simple pole at $s=1$ (implied by finiteness of class number ...). Hence $d=1$ and $L=K$.






share|cite|improve this answer









$endgroup$












  • $begingroup$
    Good point! That's certainly simpler than the adelic proof.
    $endgroup$
    – Alison Miller
    9 hours ago













11












11








11





$begingroup$

I mention this as an answer since it is too long for comments. I do not know what the adelic proof assumes. Suppose that all but finitely many primes of $K$ split completely in $L$. Suppose $d$ is the degree of $L$ over $K$. Then the zeta function of $L$ is the $d$-th power of the zeta function of $K$, up to finitely many factors. But the Zeta functions of $L$ and $K$ have only a simple pole at $s=1$ (implied by finiteness of class number ...). Hence $d=1$ and $L=K$.






share|cite|improve this answer









$endgroup$



I mention this as an answer since it is too long for comments. I do not know what the adelic proof assumes. Suppose that all but finitely many primes of $K$ split completely in $L$. Suppose $d$ is the degree of $L$ over $K$. Then the zeta function of $L$ is the $d$-th power of the zeta function of $K$, up to finitely many factors. But the Zeta functions of $L$ and $K$ have only a simple pole at $s=1$ (implied by finiteness of class number ...). Hence $d=1$ and $L=K$.







share|cite|improve this answer












share|cite|improve this answer



share|cite|improve this answer










answered 10 hours ago









VenkataramanaVenkataramana

9,09412951




9,09412951











  • $begingroup$
    Good point! That's certainly simpler than the adelic proof.
    $endgroup$
    – Alison Miller
    9 hours ago
















  • $begingroup$
    Good point! That's certainly simpler than the adelic proof.
    $endgroup$
    – Alison Miller
    9 hours ago















$begingroup$
Good point! That's certainly simpler than the adelic proof.
$endgroup$
– Alison Miller
9 hours ago




$begingroup$
Good point! That's certainly simpler than the adelic proof.
$endgroup$
– Alison Miller
9 hours ago

















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