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forum.math.uoa.gr • Προβολή θέματος - Σώμα χαρακτηριστικής p

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 Θέμα δημοσίευσης: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 12 Σεπ 2016, 18:54 
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Εγγραφή: 24 Σεπ 2014, 17:51
Δημοσ.: 48
Παιδιά , γιατί σε σώμα χαρακτηριστικής p (p διάφορος του 0 ) ισχύει ότι [unparseable or potentially dangerous latex formula];


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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 12 Σεπ 2016, 19:27 
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Εγγραφή: 11 Δεκ 2014, 19:45
Δημοσ.: 48
Ας είναι F ένα πεπερασμένο σώμα με p στοιχεία και [unparseable or potentially dangerous latex formula]

Αν [unparseable or potentially dangerous latex formula], τότε

[unparseable or potentially dangerous latex formula] (γιατί;).

Για κάθε [unparseable or potentially dangerous latex formula] ισχύει ότι [unparseable or potentially dangerous latex formula] (γιατί;). Άρα,

[unparseable or potentially dangerous latex formula]

όπως θέλαμε.

_________________
"Problems worthy of attack prove their worth by fighting back" -Paul Erdős


Τελευταία επεξεργασία απο Βασίλης4 την 12 Σεπ 2016, 20:03, επεξεργάστηκε 1 φορές συνολικά.

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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 12 Σεπ 2016, 19:43 
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Εγγραφή: 24 Σεπ 2014, 17:51
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Γιατί ισχύει το [unparseable or potentially dangerous latex formula] ;


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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 12 Σεπ 2016, 20:04 
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Εγγραφή: 11 Δεκ 2014, 19:45
Δημοσ.: 48
Άλλαξα ελαφρώς την απάντησή μου, είχα στο μυαλό μου τη περίπτωση όπου το F είναι το [unparseable or potentially dangerous latex formula]

_________________
"Problems worthy of attack prove their worth by fighting back" -Paul Erdős


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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 12 Σεπ 2016, 20:11 
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Εγγραφή: 24 Σεπ 2014, 17:51
Δημοσ.: 48
Αν το F έχει παραπάνω απο p στοιχεία ( μπορεί και άπειρο ) συνεχίζει να ισχύει ;


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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 12 Σεπ 2016, 20:31 
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Εγγραφή: 11 Δεκ 2014, 19:45
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Αν υποθέσουμε ότι το σώμα F είναι πεπερασμένο με [unparseable or potentially dangerous latex formula] στοιχεία για κάποιο θετικό ακέραιο k,

τότε θα έχουμε [unparseable or potentially dangerous latex formula], για κάθε [unparseable or potentially dangerous latex formula] καθότι θα συνεχίζουν να ισχύουν τα εξείς:

1. Για κάθε [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula] και
2. Για κάθε [unparseable or potentially dangerous latex formula], [unparseable or potentially dangerous latex formula].

Τώρα για την άπειρη περίπτωση δε το βλέπω αυτή τη στιγμή.

_________________
"Problems worthy of attack prove their worth by fighting back" -Paul Erdős


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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 15 Σεπ 2016, 20:33 
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Εγγραφή: 10 Μαρ 2016, 14:08
Δημοσ.: 32
Γεια σας.

Αν το [unparseable or potentially dangerous latex formula] είναι πεπερασμένο με τάξη [unparseable or potentially dangerous latex formula] , τότε, η πολλαπλασιαστική

ομάδα [unparseable or potentially dangerous latex formula] έχει τάξη [unparseable or potentially dangerous latex formula] ,

οπότε, για κάθε [unparseable or potentially dangerous latex formula] ισχύει [unparseable or potentially dangerous latex formula] .

Στην άπειρη περίπτωση : Έστω η άπειρη ακέραια περιοχή [unparseable or potentially dangerous latex formula] η οποία έχει

χαρακτηριστική [unparseable or potentially dangerous latex formula] . Θεωρούμε το σώμα κλασμάτων αυτής,

[unparseable or potentially dangerous latex formula]

το οποίο είναι άπειρο και έχει χαρακτηριστική [unparseable or potentially dangerous latex formula] .

Έστω [unparseable or potentially dangerous latex formula] .

Τότε, [unparseable or potentially dangerous latex formula] ενώ, [unparseable or potentially dangerous latex formula] .

Ελπίζω να μην μου ξέφυγε κάτι.


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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 16 Σεπ 2016, 11:37 
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Εγγραφή: 11 Δεκ 2014, 19:45
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Γειά σου Βαγγέλη,

καλό μου φαίνεται εμένα και μπράβο για το αντιπαράδειγμα!

_________________
"Problems worthy of attack prove their worth by fighting back" -Paul Erdős


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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 17 Σεπ 2016, 10:37 
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Εγγραφή: 24 Σεπ 2014, 17:51
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Ναι όντως απλό και ωραίο το αντιπαράδειγμα !

Και γω ήμουν σίγουρος ότι δεν πρέπει να ισχύει γιατί αν έχουμε [unparseable or potentially dangerous latex formula] θα πρέπει υποχρεωτικά

να έχουμε και [unparseable or potentially dangerous latex formula] δηλαδή το πολυώνυμο [unparseable or potentially dangerous latex formula] θα έχει άπειρες ρίζες

όταν το σώμα είναι άπειρο , πράγμα που δε μπορεί να συμβεί. ( καλά δε τα λέω ; )

Υ.Γ: Έκανα την ερώτηση γιατί έχω ένα φυλλάδιο με ασκήσεις της θεωρίας Galois και μια άσκηση έλεγε να δείξουμε ότι σε

σώμα χαρακτηριστικής p ισχύει το [unparseable or potentially dangerous latex formula].


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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 17 Σεπ 2016, 11:34 
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Εγγραφή: 09 Φεβ 2012, 22:03
Δημοσ.: 619
Στο αντί - παράδειγμα, γιατί δεν ισχύει η ισότητα;


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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 17 Σεπ 2016, 13:16 
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Εγγραφή: 11 Δεκ 2014, 19:45
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Για τον ίδιο λόγο που [unparseable or potentially dangerous latex formula] σε ένα πεπερασμένο σώμα με δυο στοιχεία.

_________________
"Problems worthy of attack prove their worth by fighting back" -Paul Erdős


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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 17 Σεπ 2016, 14:05 
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Εγγραφή: 10 Μαρ 2016, 14:08
Δημοσ.: 32
Altair έγραψε:
Στο αντί - παράδειγμα, γιατί δεν ισχύει η ισότητα;


Επειδή το [unparseable or potentially dangerous latex formula] είναι η ακολουθία

[unparseable or potentially dangerous latex formula]

ενώ το [unparseable or potentially dangerous latex formula] είναι η ακολουθία

[unparseable or potentially dangerous latex formula]

και [unparseable or potentially dangerous latex formula] στο σώμα [unparseable or potentially dangerous latex formula] .


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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 17 Σεπ 2016, 16:34 
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Εγγραφή: 09 Φεβ 2012, 22:03
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Το x δεν παίρνει τιμές 0 ή 1; (διαφορετικά, κάτι δεν κατάλαβα)


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 Θέμα δημοσίευσης: Re: Σώμα χαρακτηριστικής p
ΔημοσίευσηΔημοσιεύτηκε: 17 Σεπ 2016, 16:43 
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Εγγραφή: 11 Δεκ 2014, 19:45
Δημοσ.: 48
Όχι, το σώμα F που όρισε ο Βαγγέλης έχει ως στοιχεία του ρητές πολυωνυμικές συναρτήσεις με συντελεστές από το \mathbb{Z}/2\mathbb{Z}.

Έτσι, το x θα είναι απλά η ακολουθία [unparseable or potentially dangerous latex formula] στο [unparseable or potentially dangerous latex formula] και δεν μπορεί να είναι το ίδιο με το [unparseable or potentially dangerous latex formula], δηλαδή την ακολουθία [unparseable or potentially dangerous latex formula] στο [unparseable or potentially dangerous latex formula].

_________________
"Problems worthy of attack prove their worth by fighting back" -Paul Erdős


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