Learn Hardcore Icelandic: Kennarinn segir að góð endurtekning sé lykill að árangri. - The teacher says that good repetition is the key to success.

The basic noun is kennari (a teacher).
To say the teacher in the nominative singular, you add the masculine definite ending -inn:
- kennari → kennarinn (the teacher)

This is very common:

bók → bókin (book → the book)
stóll → stóllinn (chair → the chair)

So Kennarinn (capitalized only because it’s at the start of the sentence) literally means the-teacher in one word.

What tense and person is segir, and what is the infinitive form of this verb?

Segir is:

present tense
3rd person singular
indicative mood

The infinitive form is að segja (to say).

A small present-tense paradigm:

ég segi – I say
þú segir – you (sg.) say
hann / hún segir – he / she says
við segjum – we say
þið segið – you (pl.) say
þeir / þær / þau segja – they say

So Kennarinn segir = The teacher says.

There are two að in the sentence. Do they mean the same thing?

No, they are two different functions of að:

að after segir:
- segir að góð endurtekning sé…
  Here að is a conjunction (often called a “complementizer”), meaning that.
- It introduces a subordinate clause: að góð endurtekning sé lykill að árangri.
að before árangri:
- lykill að árangri
  Here að is a preposition, roughly to / toward / for in this expression, giving key to success.

So:

first að = that (linking a clause)
second að = to (linking a noun phrase and taking the dative case)

Why is it sé and not er for is in the clause að góð endurtekning sé lykill að árangri?

Sé is the present subjunctive of vera (to be).

Icelandic often uses the subjunctive in reported speech or indirect statements after verbs like segja (to say), halda (to think), trúa (to believe), etc.

er = present indicative (plain is)
sé = present subjunctive (used here because it’s what someone says)

So:

Kennarinn segir að góð endurtekning sé lykill að árangri.
Literally: The teacher says that good repetition be the key to success → idiomatically: …is the key…

In everyday speech, many Icelanders also use er here:

Kennarinn segir að góð endurtekning er lykill að árangri.

But the subjunctive (sé) is the “textbook” choice after segir að… because it marks this as the content of someone’s words, not a bare factual assertion by the narrator.

How does adjective agreement work in góð endurtekning?

In Icelandic, adjectives agree with the noun they describe in:

gender
number
case
and also definiteness (strong vs. weak declension)

Here:

endurtekning (repetition) is feminine, singular, nominative, and indefinite.
The adjective is góður (good) in its strong declension.

The form that matches a feminine singular nominative indefinite noun is:

góð

So:

góð endurtekning = good repetition
(fem. sg. nom. adj. + fem. sg. nom. noun)

If the noun were masculine or neuter, the form would change, e.g.:

góður kennari – good teacher (masc. nom. sg.)
gott barn – good child (neut. nom. sg.)

Which words are in the nominative case here, and why?

The nominative is used mainly for subjects and for predicative complements (after vera, to be).

In this sentence:

Kennarinn
- Subject of the main verb segir
- Therefore nominative.
Inside the subordinate clause að góð endurtekning sé lykill að árangri:
- endurtekning (in góð endurtekning) is the subject of sé, so it’s nominative.
- lykill is the predicative complement of sé (what the subject “is”), and in Icelandic those usually match the subject’s case, so lykill is also nominative.
árangri is not nominative; it’s dative because of the preposition að.

So nominative forms here are:

Kennarinn, endurtekning, lykill

Why is árangri in this form and not árangur?

The dictionary form árangur is nominative singular.

In the phrase lykill að árangri, the noun is governed by the preposition að, which (in this meaning) takes the dative case.

The noun árangur declines (simplified):

nominative: árangur
accusative: árangur
dative: árangri
genitive: árangurs

Because að here requires dative, you must use:

árangri (dative singular)

So lykill að árangri literally is key to/toward success, with árangri in the dative.

Why is it lykill að árangri and not something like lykill af árangri or lykill árangurs?

The expression lykill að e-u (key to something) is a set phrase in Icelandic.

lykill að árangri – the idiomatic way to say key to success
að in this sense (“leading to, towards”) commonly selects the dative.

Other options would sound wrong or at least very odd here:

lykill af árangri – af is more like from / of / made of; this doesn’t match the English idea of a key that leads to success.
lykill árangurs – genitive árangurs would be something like key of success, which is not how Icelandic normally phrases this relationship.

So you should memorize lykill að e-u as the natural pattern:
lykill að árangri, lykill að hamingju, etc.

In English we say the key to success, with the. Why is there no definite article on lykill or endurtekning here?

English and Icelandic don’t match perfectly in when they use definite vs. indefinite forms for general statements.

In this sentence we are talking about a general truth, not one specific repetition or one specific key:

góð endurtekning – good repetition (in general)
lykill að árangri – a key to success / the key to success (in general)

Icelandic often uses the indefinite for such generalizations:

Góð endurtekning er lykill að árangri.
Reykingar eru slæmar fyrir heilsuna. – Smoking is bad for your health.

If you made endurtekningin / lykillinn definite, it would sound like you’re talking about a specific, previously mentioned kind of repetition or key:

Góða endurtekningin er lykillinn að árangrinum.
→ something more like “That good repetition is the key to that success” (much more specific and contextual).

So the Icelandic indefinite here corresponds to an English generic (“the key to success” as a general idea).

Could you rearrange the clause að góð endurtekning sé lykill að árangri and still be correct, or is the word order fixed?

The word order here is the neutral, most natural one:

[að] [góð endurtekning] [sé] [lykill að árangri]
(that) (good repetition) (is) (key to success)

In standard usage:

The subject (góð endurtekning) normally comes before the verb in this kind of að-clause.
The complement (lykill að árangri) follows the verb.

You can move parts around for emphasis or stylistic effect, for example:

Kennarinn segir að lykill að árangri sé góð endurtekning.
(focusing first on “the key to success”)

That is still grammatical, but has a slightly different emphasis.

Highly marked orders like:

að sé góð endurtekning lykill að árangri

would be unusual or feel literary/odd. As a learner, it’s safest to stick to:

að [subject] [verb] [complement]
→ að góð endurtekning sé lykill að árangri

Why is there no comma before að, unlike English The teacher says that good repetition…?

Icelandic comma rules are different from English:

English often puts a comma before that-clauses in certain styles:
The teacher says that… (sometimes with or without a comma, depending on style).
Icelandic normally does not put a comma before að when it introduces a complement clause after verbs like segja, vita, halda, etc.

So:

Kennarinn segir að góð endurtekning sé lykill að árangri.
(no comma)

You might see commas used in other structures (e.g. some relative clauses, lists, parenthetical clauses), but not routinely before this kind of að.