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From Travian v2 Wiki (S2-S6), the encyclopedia anybody can edit

The Blacksmith is the building in which you can upgrade your troops' offense.

The level of your blacksmith dictates the level of the research you may undertake, ie with a level 2 blacksmith you can upgrade any of your troops' offense up to level 2. To upgrade them to level 5 you would need a level 5 blacksmith.

How do upgrades work? Each upgrade level adds a cummulative bonus of 1.5% to the attack rating of a unit. Here is an explanation of how upgrades would work with a Roman Legionnaire based on a post by Mon-K on the Travian Forum:

For example, your Roman Legionnaire will have:

40 without any Blacksmith upgrades.

40.6 with the Level 1 upgrade.

41.2 with the Level 2 upgrade.

And so on.

Each time an upgrade is completed it is granted instantly to all past and future troops (whether they are in the village or on an attack, returning, training, etc.) These bonuses are especially useful when you have large numbers (100+) of troops.

Costs and time to build level 1:

 170 |  200 |  380 |  130 |  4 |   ?

Requirements:
Academy Level 3

Level							Culture Points
1	170	200	380	130	?	4	2
2	220	255	485	165	?	2	3
3	280	330	625	215	?	2	3
4	355	420	795	275	?	2	4
5	455	535	1020	350	?	2	5
6	750	880	1670	570	?	3	6
7	955	1125	2140	730	?	3	7
8	955	1125	2140	730	?	3	9
9	1225	1440	2740	935	?	3	10
10	1570	1845	3505	1200	?	3	12
11	2005	2360	4485	1535	?	3	15
12	2570	3020	4740	1965	?	3	18
13	3290	3870	7350	2515	?	3	21
14	4210	4950	9410	3220	?	3	26
15	5390	6340	12045	4120	?	3	31
16	6895	8115	15415	5275	?	4	37
17	8825	10385	19730	6750	?	4	44
18	11300	13290	25255	8640	?	4	53
19	14460	17015	32325	11060	?	4	64
20	18510	21780	41380	14155	?	4	77

Construction Times

Construction Period Based on Main Building (MB) Level - Levels 1-10

Blacksmith Level	MB Level 1	MB Level 2	MB Level 3	MB Level 4	MB Level 5	MB Level 6	MB Level 7	MB Level 8	MB Level 9	MB Level 10
1	-	-	0:33:20	0:33:20	0:33:20	0:33:20	0:33:20	0:33:20	0:33:20	0:33:20
2	-	-	0:40:30	0:39:10	0:37:40	0:36:20	0:35:00	0:33:50	0:32:30	0:31:20
3	-	-	0:51:40	0:49:50	0:48:00	0:46:20	0:44:40	0:43:00	0:41:30	0:40:00
4	-	-	1:04:40	1:02:20	1:00:00	0:57:50	0:55:50	0:53:50	0:51:50	0:50:00
5	-	-	1:19:40	1:16:50	1:14:00	1:11:20	1:08:50	1:06:20	1:03:50	1:01:40
6	-	-	1:37:00	1:33:30	1:30:10	1:26:50	1:23:50	1:20:50	1:17:50	1:15:00
7	-	-	1:57:10	1:53:00	1:48:50	1:45:00	1:41:10	1:37:30	1:34:00	1:30:40
8	-	-	2:20:30	2:15:30	2:10:40	2:06:00	2:01:20	1:57:00	1:52:50	1:48:50
9	-	-	2:47:40	2:41:40	2:35:50	2:30:20	2:24:50	2:19:40	2:14:40	2:09:50
10	-	-	3:19:10	3:12:00	3:05:10	2:58:30	2:52:00	2:45:50	2:39:50	2:34:10
11	-	-	3:55:40	3:47:10	3:39:00	3:31:10	3:23:30	3:16:10	3:09:10	3:02:20
12	-	-	4:38:00	4:28:00	4:18:30	4:09:10	4:00:10	3:51:30	3:43:10	3:35:10
13	-	-	5:27:10	5:15:30	5:04:10	4:53:10	4:42:40	4:32:20	4:22:40	4:13:10
14	-	-	6:24:10	6:10:20	5:57:00	5:44:10	5:31:50	5:19:50	5:08:20	4:57:20
15	-	-	7:30:20	7:14:10	6:58:30	6:43:30	6:28:50	6:14:50	6:01:30	5:48:20
16	-	-	8:47:00	8:28:00	8:09:50	7:52:10	7:35:10	7:18:50	7:03:00	6:47:40
17	-	-	10:16:00	9:53:50	9:32:30	9:11:50	8:52:00	8:32:50	8:14:20	7:56:30
18	-	-	11:59:10	11:33:20	11:08:20	10:44:20	10:21:10	9:58:50	9:37:10	9:16:30
19	-	-	13:59:00	13:28:50	12:59:40	12:31:30	12:04:30	11:38:30	11:13:20	10:49:00
20	-	-	16:17:50	15:42:40	15:08:40	14:36:00	14:04:30	13:34:00	13:04:40	12:36:30

Construction Period Based on Main Building (MB) Level - Levels 11-20

Blacksmith Level	MB Level 11	MB Level 12	MB Level 13	MB Level 14	MB Level 15	MB Level 16	MB Level 17	MB Level 18	MB Level 19	MB Level 20
1	0:33:20	0:33:20	0:33:20	0:33:20	0:33:20	0:33:20	0:33:20	0:33:20	0:33:20	0:33:20
2	0:30:20	0:29:10	0:28:10	0:27:10	0:26:10	0:25:10	0:24:20	0:23:20	0:22:30	0:21:50
3	0:38:30	0:37:10	0:35:50	0:34:30	0:33:20	0:32:10	0:31:00	0:29:50	0:28:50	0:27:40
4	0:48:10	0:46:30	0:44:50	0:43:10	0:41:40	0:40:10	0:38:40	0:37:20	0:36:00	0:34:40
5	0:59:20	0:57:10	0:55:10	0:53:10	0:51:20	0:49:30	0:47:40	0:46:00	0:44:20	0:42:40
6	1:12:20	1:09:40	1:07:10	1:04:50	1:02:30	1:00:10	0:58:00	0:56:00	0:54:00	0:52:00
7	1:27:20	1:24:10	1:21:10	1:18:20	1:15:30	1:12:50	1:10:10	1:07:40	1:05:10	1:02:50
8	1:44:50	1:41:00	1:37:30	1:34:00	1:30:30	1:27:20	1:24:10	1:21:10	1:18:10	1:15:20
9	2:05:10	2:00:30	1:56:10	1:52:00	1:48:00	1:44:10	1:40:20	1:36:50	1:33:20	1:30:00
10	2:28:30	2:23:10	2:18:00	2:13:10	2:08:20	2:03:40	1:59:10	1:55:00	1:50:50	1:46:50
11	2:55:50	2:49:30	2:43:20	2:37:30	2:31:50	2:26:20	2:21:10	2:16:00	2:11:10	2:06:20
12	3:27:20	3:20:00	3:12:40	3:05:50	2:59:10	2:52:40	2:46:30	2:40:30	2:34:40	2:29:10
13	4:04:00	3:55:20	3:46:50	3:38:40	3:30:40	3:23:10	3:15:50	3:08:50	3:02:00	2:55:30
14	4:46:30	4:36:10	4:26:20	4:16:40	4:07:30	3:58:30	3:50:00	3:41:40	3:33:40	3:26:00
15	5:35:50	5:23:50	5:12:10	5:00:50	4:50:00	4:39:40	4:29:30	4:19:50	4:10:30	4:01:30
16	6:33:00	6:19:00	6:05:20	5:52:10	5:39:30	5:27:10	5:15:30	5:04:10	4:53:10	4:42:40
17	7:39:30	7:22:50	7:07:00	6:51:30	6:36:50	6:22:30	6:08:40	5:55:30	5:42:40	5:30:20
18	8:56:20	8:37:10	8:18:30	8:00:30	7:43:10	7:26:30	7:10:30	6:55:00	6:40:00	6:25:40
19	10:25:40	10:03:10	9:41:30	9:20:30	9:00:20	8:40:50	8:22:10	8:04:00	7:46:40	7:29:50
20	12:09:20	11:43:00	11:17:40	10:53:20	10:29:50	10:07:10	9:45:20	9:24:10	9:03:50	8:44:20

Is there any problem with the first level of the Blacksmith(time)? No, you need Main Building level 3 to build Blacksmith level 1